1. HCF and LCM at a Glance
HCF: Greatest positive integer that divides all the given positive integers exactly.
LCM: Smallest positive integer exactly divisible by all the given positive integers.
HCF is also called GCD.
2. Factor and Multiple
Factor: Divides a number exactly.
Multiple: Obtained by multiplying a number by an integer.
Exam Tip:
HCF is based on common factors, whereas LCM is based on common multiples.
3. Basic Size Rule
For positive integers a and b:
HCF(a,b) ≤ smaller number
LCM(a,b) ≥ larger number
4. When One Number Divides the Other
If a divides b exactly and a≤b:
HCF(a,b)=a
LCM(a,b)=b
5. Co-prime Numbers
Two positive integers are co-prime when their HCF is 1.
Example:
8 and 15 are co-prime although neither condition requires both numbers to be prime.
6. LCM of Two Co-prime Numbers
If HCF(a,b)=1, then:
LCM(a,b)=ab
7. Consecutive Numbers
Any two consecutive positive integers are co-prime.
Therefore:
HCF=1
LCM=their product
8. Equal Numbers and Number 1
HCF(a,a)=a
LCM(a,a)=a
HCF(1,n)=1
LCM(1,n)=n
9. Fundamental Product Relation
For exactly two positive integers:
First Number × Second Number = HCF × LCM
Exam Trap:
Do not directly extend this simple identity to three or more numbers.
10. Finding an Unknown Number
Unknown Number = (HCF × LCM) ÷ Known Number
11. HCF Must Divide the LCM
For valid positive integers:
LCM must be exactly divisible by HCF.
12. Standard HCF Representation
If HCF=h, write the two numbers as:
hm and hn,
where HCF(m,n)=1.
13. LCM in HCF Representation
If numbers are hm and hn with HCF(m,n)=1:
LCM=hmn
14. Common Scaling Property
HCF(ka,kb)=k×HCF(a,b)
LCM(ka,kb)=k×LCM(a,b)
15. Prime Factorisation Rule for HCF
HCF = Common prime factors taken with their lowest powers.
Memory Rule:
HCF → Common + Lowest
16. Prime Factorisation Rule for LCM
LCM = All required prime factors taken with their highest powers.
Memory Rule:
LCM → All + Highest
17. Ladder Method for HCF
In the HCF ladder method, divide all current numbers together only by a factor common to every one of them.
Multiply all common divisors used.
18. Ladder Method for LCM
In the LCM ladder method, a prime divisor may divide one or more current numbers.
Numbers not divisible by it are carried forward unchanged.
19. Successive Division Method
For two numbers:
Larger ÷ Smaller → remainder
Previous divisor ÷ remainder → next remainder
Continue until remainder becomes 0.
Last non-zero divisor = HCF.
20. Euclidean Division Identity
Dividend = Divisor × Quotient + Remainder
21. Reverse Successive Division
If successive quotients and the final HCF are given, reconstruct the numbers backwards using:
Dividend = Divisor × Quotient + Remainder
Exam Tip:
Do not simply multiply the given quotients.
22. Stepwise HCF of Several Numbers
HCF(a,b,c)=HCF(HCF(a,b),c)
23. Stepwise LCM of Several Numbers
LCM(a,b,c)=LCM(LCM(a,b),c)
24. HCF of Fractions
Under the usual competitive-exam convention for positive fractions:
HCF of fractions
= HCF of numerators ÷ LCM of denominators
Exam Tip:
Reduce fractions to lowest terms where possible before calculation.
25. LCM of Fractions
LCM of fractions
= LCM of numerators ÷ HCF of denominators
Memory Rule:
HCF of fractions → HCF above, LCM below.
LCM of fractions → LCM above, HCF below.
26. Mixed Fractions
Convert every mixed fraction into an improper fraction before finding HCF or LCM.
Common Mistake:
Do not separately work with the whole-number and fractional parts.
27. Decimals — Common Scaling Rule
For terminating decimals:
1. Make decimal places equal.
2. Multiply all values by the same power of 10.
3. Find integer HCF or LCM.
4. Divide the answer by the same scaling factor.
28. Fractions and Decimals Together
Convert all quantities into one consistent exact form before calculation.
Shortcut:
Use whichever exact representation produces the simplest integers.
29. Reciprocal Relationship
Under the usual positive-rational aptitude convention:
HCF of reciprocals = reciprocal of original LCM
LCM of reciprocals = reciprocal of original HCF
30. Greatest Equal Measurement
Greatest possible equal measuring unit = HCF of the measurements.
31. Largest Equal Pieces
Maximum possible length or size of identical pieces = HCF.
32. Largest Square Tile
For an integral rectangular floor after unit conversion:
Side of largest square tile = HCF(length,breadth)
Number of Tiles:
(length ÷ tile side) × (breadth ÷ tile side)
33. Maximum Identical Groups
Maximum number of identical groups = HCF of the quantities.
Contents per Group:
Divide each original quantity by the HCF.
34. Repeated Events
If several events repeat at fixed intervals:
Time after which all occur together again = LCM of the intervals.
35. Counting Simultaneous Occurrences
If events start together at time 0 and both endpoints are included:
Number of common occurrences
= floor(T ÷ LCM) + 1
Exam Trap:
Add 1 only when the starting occurrence is actually included.
36. Least Number Divisible by Several Numbers
Least positive number exactly divisible by several positive integers = their LCM.
37. Least Addition for Common Divisibility
Let L=LCM and N leave remainder r on division by L.
If r≠0:
Least number to add = L−r
38. Least Subtraction for Common Divisibility
If N leaves remainder r when divided by the required LCM:
Least number to subtract = r
39. Known Same Remainder — HCF
Greatest divisor leaving known remainder r from a,b,c:
HCF(a−r,b−r,c−r)
40. Unknown Same Remainder — HCF
If the same remainder is left but its value is unknown:
Take the HCF of pairwise differences.
For a,b,c:
Use differences such as b−a, c−b and c−a.
41. Different Remainders — HCF
If a,b,c leave remainders r₁,r₂,r₃ respectively:
Required greatest divisor
=HCF(a−r₁,b−r₂,c−r₃)
Necessary Check:
The divisor must be greater than every stated remainder.
42. Same-Remainder LCM Form
If N leaves remainder r when divided by a,b,c:
N−r is divisible by all of them.
Therefore:
N=k×LCM(a,b,c)+r
Important:
Check the lower-bound and additional conditions in the question before declaring a particular value the least solution.
43. Ratio and HCF
If the reduced ratio is m:n and HCF=h:
Numbers=hm and hn
LCM=hmn
Exam Trap:
Always reduce the ratio first.
44. Ratio and LCM
If reduced ratio=m:n and LCM=L:
HCF=L/(mn)
45. Possible Number Pairs from HCF and LCM
If HCF=h and LCM=L:
mn=L/h
Find factor pairs of L/h and retain only co-prime pairs.
Original numbers=hm and hn.
46. Counting Possible Pairs
Let N=LCM/HCF.
Count unordered co-prime factor pairs of N.
Shortcut:
If N>1 has k distinct prime factors:
Number of unordered pairs=2k−1.
47. Sum and Difference Conditions
If numbers=hm and hn:
Sum S → m+n=S/h
Difference D → |m−n|=D/h
48. HCF, LCM and Sum Together
If HCF=h, LCM=L and sum=S:
mn=L/h
m+n=S/h
Useful Equation:
t2−(S/h)t+(L/h)=0
49. Derived Relationships
For two positive numbers a and b:
ab=HCF×LCM
Mean proportional=√(HCF×LCM)
If a+b=S:
1/a + 1/b
=S/(HCF×LCM)
50. Final 30-Second Revision Sheet
Core Formulas:
HCF×LCM = Product of two numbers
HCF of fractions = HCF(numerators)/LCM(denominators)
LCM of fractions = LCM(numerators)/HCF(denominators)
Greatest equal measure → HCF
Largest square tile → HCF
Maximum identical groups → HCF
Events together again → LCM
Least divisible number → LCM
Known same remainder r → HCF(numbers−r)
Unknown same remainder → HCF(pairwise differences)
Different remainders → HCF(number−corresponding remainder)
If HCF=h:
Numbers=hm,hn with HCF(m,n)=1
LCM=hmn
If HCF=h and LCM=L:
mn=L/h
Ratio m:n + HCF h → hm,hn
Ratio m:n + LCM L → HCF=L/(mn)
If L/H has k distinct prime factors:
Possible unordered pairs=2k−1
Final Exam Strategy:
First identify the mathematical pattern, then choose HCF or LCM, convert all units or forms consistently, and only then calculate.
Previous-Year Questions
Chapter 3 PYQ Bank — Covers fundamental HCF-LCM properties, prime factorisation, product relation, fractions, decimals, applications, remainder problems and advanced ratio-based questions.
SSC GD PYQBasic HCF24 February 2024 · Shift I
Q1. Find the HCF of 84, 126 and 168.
A. 44
B. 49
C. 40
D. 42
Correct Answer: D. 42
84=22×3×7, 126=2×32×7 and 168=23×3×7.
Using the lowest powers of common primes:
HCF=2×3×7=42.
SSC CGL PYQHCF-LCM Validity1 December 2022 · Tier-I · Shift I
Q2. The HCF of two numbers is 12. Which of the following cannot be their LCM?
A. 72
B. 60
C. 90
D. 84
Correct Answer: C. 90
The LCM must be exactly divisible by the HCF.
72, 60 and 84 are divisible by 12, but 90 is not.
Therefore 90 cannot be the LCM.
SSC GD PYQHCF : LCM12 February 2019 · Shift II
Q3. What is the ratio of the HCF to the LCM of 16 and 20?
A. 1 : 25
B. 4 : 63
C. 1 : 20
D. 1 : 10
Correct Answer: C. 1 : 20
HCF(16,20)=4 and LCM(16,20)=80.
Required ratio=4:80=1:20.
SSC MTS PYQProduct Relation13 July 2022 · Shift I
Q4. The product of two numbers is 432 and their HCF is 6. Find their LCM.
A. 68
B. 48
C. 78
D. 72
Correct Answer: D. 72
Product=HCF×LCM.
LCM=432÷6=72.
SSC MTS PYQProduct Relation4 May 2023 · Shift II
Q5. The LCM of two numbers is 72 and their product is 1728. Find their HCF.
A. 1
B. 18
C. 24
D. 12
Correct Answer: C. 24
HCF=Product÷LCM
=1728÷72
=24.
SSC MTS PYQDerived Relation26 October 2021 · Shift I
Q6. The HCF and LCM of two numbers are 12 and 48 respectively. What is the square root of the product of the numbers?
A. 16
B. 24
C. 48
D. 12
Correct Answer: B. 24
Product=12×48=576.
√576=24.
SSC GD PYQHCF of Fractions13 January 2023 · Shift III
Q7. Find the HCF of 3/8, 5/12 and 15/16.
A. 1/12
B. 1/24
C. 1/48
D. 1/36
Correct Answer: C. 1/48
HCF of numerators=HCF(3,5,15)=1.
LCM of denominators=LCM(8,12,16)=48.
HCF=1/48.
SSC MTS PYQLCM of Fractions9 October 2024 · Shift I
Q8. Find the LCM of 1/7, 2/5 and 3/15.
A. 6
B. 4
C. 5
D. 3
Correct Answer: A. 6
LCM of numerators=LCM(1,2,3)=6.
HCF of denominators=HCF(7,5,15)=1.
Required LCM=6.
RRB NTPC PYQLCM of Decimals15 June 2026 · UG CBT-I · Shift III
Q9. Find the LCM of 5, 3.2 and 0.03.
A. 215
B. 185
C. 120
D. 240
Correct Answer: D. 240
Write 5.00, 3.20 and 0.03 and multiply all by100:
500,320,3.
Their LCM=24000.
Required LCM=24000÷100=240.
RRB NTPC PYQLCM of Decimals7 August 2025 · UG CBT-I · Shift I
Q10. Find the LCM of 15.4 and 0.006.
A. 0.462
B. 462
C. 46.2
D. 4.62
Correct Answer: C. 46.2
15.4=77/5 and 0.006=3/500.
LCM of numerators=231.
HCF of denominators=5.
LCM=231/5=46.2.
SSC GD PYQGreatest Measurement24 November 2021 · Shift I
Q11. What is the greatest possible length that can exactly measure 4 m 90 cm, 3 m 85 cm and 12 m 95 cm?
A. 35 cm
B. 25 cm
C. 20 cm
D. 45 cm
Correct Answer: A. 35 cm
Convert into centimetres:
490,385,1295.
HCF(490,385,1295)=35 cm.
SSC GD PYQBells & Repeated Events31 January 2023 · Shift I
Q12. Three bells start together and ring every 8 sec, 10 sec and 12 sec. How many times do they ring together in 30 minutes, including the starting instant?
A. 14
B. 17
C. 16
D. 18
Correct Answer: C. 16
LCM(8,10,12)=120 sec.
30 min=1800 sec.
1800÷120=15 common intervals.
Including the initial ringing:
15+1=16.
SSC GD PYQSame Remainder26 November 2021 · Shift III
Q13. Find the greatest number that leaves the same remainder when dividing 265, 580 and 825.
A. 35
B. 55
C. 25
D. 45
Correct Answer: A. 35
Use pairwise differences:
580−265=315
825−580=245
825−265=560.
HCF(315,245,560)=35.
SSC MTS PYQSame Remainder & LCM4 November 2024 · Shift II
Q14. Find the smallest number that leaves remainder 6 when divided by 20, 15 and 30.
A. 36
B. 66
C. 56
D. 96
Correct Answer: B. 66
LCM(20,15,30)=60.
Required number=60+6=66.
SSC GD PYQLeast Number3 March 2019 · Shift III
Q15. Find the least positive number which becomes exactly divisible by 4, 5, 6 and 7 after 8 is added to it.
A. 322
B. 312
C. 412
D. 422
Correct Answer: C. 412
LCM(4,5,6,7)=420.
Required number=420−8=412.
SSC CGL PYQRatio & HCF1 December 2022 · Tier-I · Shift III
Q16. Two numbers are in the ratio 6:5 and their HCF is 3. Find their LCM.
A. 64
B. 110
C. 90
D. 80
Correct Answer: C. 90
6 and5 are co-prime.
Numbers=18 and15.
LCM=3×6×5=90.
SSC CGL PYQRatio & HCF13 December 2022 · Tier-I · Shift I
Q17. The HCF of two numbers is 9 and their ratio is 14:19. Find their LCM.
A. 2394
B. 3990
C. 1596
D. 3192
Correct Answer: A. 2394
14 and19 are co-prime.
LCM=9×14×19=2394.
SSC MTS PYQHCF, LCM & Ratio20 July 2022 · Shift III
Q18. The HCF and LCM of two numbers are 35 and 840 respectively. If their ratio is 3:8, find the smaller number.
A. 35
B. 280
C. 105
D. 210
Correct Answer: C. 105
Since3 and8 are co-prime, the numbers are:
35×3=105 and35×8=280.
Smaller number=105.
SSC MTS PYQHCF, LCM & Difference27 October 2021 · Shift I
Q19. The HCF and LCM of two numbers are 6 and 90 respectively. If one number exceeds the other by 12, find the greater number.
A. 12
B. 45
C. 30
D. 51
Correct Answer: C. 30
Product=6×90=540.
The pair with product540 and difference12 is18 and30.
Greater number=30.
RRB Group D PYQAdvanced HCF-LCM1 September 2022 · Shift III
Q20. Two numbers are in the ratio 4:3, and the product of their HCF and LCM is 432. Find the sum of the reciprocals of their HCF and LCM.
A. 13/72
B. 13/27
C. 13/2
D. 13/7
Correct Answer: A. 13/72
Let the numbers be4x and3x.
Their product=12x²=432, so x=6.
Numbers=24 and18.
HCF=6 and LCM=72.
1/6+1/72=13/72.
SSC GD PYQBasic HCF24 February 2024 · Shift I
Q21. Find the HCF of 84, 126 and 168.
A. 44
B. 49
C. 40
D. 42
Correct Answer: D. 42
84=22×3×7, 126=2×32×7 and 168=23×3×7.
Using the lowest powers of common primes:
HCF=2×3×7=42.
SSC CGL PYQHCF-LCM Validity1 December 2022 · Tier-I · Shift I
Q22. The HCF of two numbers is 12. Which of the following cannot be their LCM?
A. 72
B. 60
C. 90
D. 84
Correct Answer: C. 90
The LCM must be exactly divisible by the HCF.
72, 60 and 84 are divisible by 12, but 90 is not.
Therefore 90 cannot be the LCM.
SSC GD PYQHCF : LCM12 February 2019 · Shift II
Q23. What is the ratio of the HCF to the LCM of 16 and 20?
A. 1 : 25
B. 4 : 63
C. 1 : 20
D. 1 : 10
Correct Answer: C. 1 : 20
HCF(16,20)=4 and LCM(16,20)=80.
Required ratio=4:80=1:20.
SSC MTS PYQProduct Relation13 July 2022 · Shift I
Q24. The product of two numbers is 432 and their HCF is 6. Find their LCM.
A. 68
B. 48
C. 78
D. 72
Correct Answer: D. 72
Product=HCF×LCM.
LCM=432÷6=72.
SSC MTS PYQProduct Relation4 May 2023 · Shift II
Q25. The LCM of two numbers is 72 and their product is 1728. Find their HCF.
A. 1
B. 18
C. 24
D. 12
Correct Answer: C. 24
HCF=Product÷LCM
=1728÷72
=24.
SSC MTS PYQDerived Relation26 October 2021 · Shift I
Q26. The HCF and LCM of two numbers are 12 and 48 respectively. What is the square root of the product of the numbers?
A. 16
B. 24
C. 48
D. 12
Correct Answer: B. 24
Product=12×48=576.
√576=24.
SSC GD PYQHCF of Fractions13 January 2023 · Shift III
Q27. Find the HCF of 3/8, 5/12 and 15/16.
A. 1/12
B. 1/24
C. 1/48
D. 1/36
Correct Answer: C. 1/48
HCF of numerators=HCF(3,5,15)=1.
LCM of denominators=LCM(8,12,16)=48.
HCF=1/48.
SSC MTS PYQLCM of Fractions9 October 2024 · Shift I
Q28. Find the LCM of 1/7, 2/5 and 3/15.
A. 6
B. 4
C. 5
D. 3
Correct Answer: A. 6
LCM of numerators=LCM(1,2,3)=6.
HCF of denominators=HCF(7,5,15)=1.
Required LCM=6.
RRB NTPC PYQLCM of Decimals15 June 2026 · UG CBT-I · Shift III
Q29. Find the LCM of 5, 3.2 and 0.03.
A. 215
B. 185
C. 120
D. 240
Correct Answer: D. 240
Write 5.00, 3.20 and 0.03 and multiply all by100:
500,320,3.
Their LCM=24000.
Required LCM=24000÷100=240.
RRB NTPC PYQLCM of Decimals7 August 2025 · UG CBT-I · Shift I
Q30. Find the LCM of 15.4 and 0.006.
A. 0.462
B. 462
C. 46.2
D. 4.62
Correct Answer: C. 46.2
15.4=77/5 and 0.006=3/500.
LCM of numerators=231.
HCF of denominators=5.
LCM=231/5=46.2.
SSC GD PYQGreatest Measurement24 November 2021 · Shift I
Q31. What is the greatest possible length that can exactly measure 4 m 90 cm, 3 m 85 cm and 12 m 95 cm?
A. 35 cm
B. 25 cm
C. 20 cm
D. 45 cm
Correct Answer: A. 35 cm
Convert into centimetres:
490,385,1295.
HCF(490,385,1295)=35 cm.
SSC GD PYQBells & Repeated Events31 January 2023 · Shift I
Q32. Three bells start together and ring every 8 sec, 10 sec and 12 sec. How many times do they ring together in 30 minutes, including the starting instant?
A. 14
B. 17
C. 16
D. 18
Correct Answer: C. 16
LCM(8,10,12)=120 sec.
30 min=1800 sec.
1800÷120=15 common intervals.
Including the initial ringing:
15+1=16.
SSC GD PYQSame Remainder26 November 2021 · Shift III
Q33. Find the greatest number that leaves the same remainder when dividing 265, 580 and 825.
A. 35
B. 55
C. 25
D. 45
Correct Answer: A. 35
Use pairwise differences:
580−265=315
825−580=245
825−265=560.
HCF(315,245,560)=35.
SSC MTS PYQSame Remainder & LCM4 November 2024 · Shift II
Q34. Find the smallest number that leaves remainder 6 when divided by 20, 15 and 30.
A. 36
B. 66
C. 56
D. 96
Correct Answer: B. 66
LCM(20,15,30)=60.
Required number=60+6=66.
SSC GD PYQLeast Number3 March 2019 · Shift III
Q35. Find the least positive number which becomes exactly divisible by 4, 5, 6 and 7 after 8 is added to it.
A. 322
B. 312
C. 412
D. 422
Correct Answer: C. 412
LCM(4,5,6,7)=420.
Required number=420−8=412.
SSC CGL PYQRatio & HCF1 December 2022 · Tier-I · Shift III
Q36. Two numbers are in the ratio 6:5 and their HCF is 3. Find their LCM.
A. 64
B. 110
C. 90
D. 80
Correct Answer: C. 90
6 and5 are co-prime.
Numbers=18 and15.
LCM=3×6×5=90.
SSC CGL PYQRatio & HCF13 December 2022 · Tier-I · Shift I
Q37. The HCF of two numbers is 9 and their ratio is 14:19. Find their LCM.
A. 2394
B. 3990
C. 1596
D. 3192
Correct Answer: A. 2394
14 and19 are co-prime.
LCM=9×14×19=2394.
SSC MTS PYQHCF, LCM & Ratio20 July 2022 · Shift III
Q38. The HCF and LCM of two numbers are 35 and 840 respectively. If their ratio is 3:8, find the smaller number.
A. 35
B. 280
C. 105
D. 210
Correct Answer: C. 105
Since3 and8 are co-prime, the numbers are:
35×3=105 and35×8=280.
Smaller number=105.
SSC MTS PYQHCF, LCM & Difference27 October 2021 · Shift I
Q39. The HCF and LCM of two numbers are 6 and 90 respectively. If one number exceeds the other by 12, find the greater number.
A. 12
B. 45
C. 30
D. 51
Correct Answer: C. 30
Product=6×90=540.
The pair with product540 and difference12 is18 and30.
Greater number=30.
RRB Group D PYQAdvanced HCF-LCM1 September 2022 · Shift III
Q40. Two numbers are in the ratio 4:3, and the product of their HCF and LCM is 432. Find the sum of the reciprocals of their HCF and LCM.
A. 13/72
B. 13/27
C. 13/2
D. 13/7
Correct Answer: A. 13/72
Let the numbers be4x and3x.
Their product=12x²=432, so x=6.
Numbers=24 and18.
HCF=6 and LCM=72.
1/6+1/72=13/72.
Practice MCQs
Questions 1–20: Basic HCF-LCM concepts, direct calculations and fundamental properties.
Practice MCQ
Q1. Find the HCF of 72 and 120.
A. 12
B. 18
C. 24
D. 36
Correct Answer: C. 24
HCF(72,120)=24. Practice MCQ
Q2. Find the LCM of 18 and 24.
A. 36
B. 48
C. 72
D. 96
Correct Answer: C. 72
18=2×3² and 24=2³×3. LCM=2³×3²=72. Practice MCQ
Q3. Find the HCF of 96 and 144.
A. 24
B. 36
C. 48
D. 72
Correct Answer: C. 48
The greatest common factor of 96 and 144 is 48. Practice MCQ
Q4. Find the LCM of 12, 15 and 20.
A. 30
B. 45
C. 60
D. 120
Correct Answer: C. 60
LCM(12,15,20)=60. Practice MCQ
Q5. Find the HCF of 84 and 126.
A. 21
B. 28
C. 42
D. 63
Correct Answer: C. 42
84=42×2 and 126=42×3. Therefore HCF=42. Practice MCQ
Q6. Find the LCM of 21, 28 and 36.
A. 126
B. 168
C. 252
D. 504
Correct Answer: C. 252
LCM(21,28,36)=252. Practice MCQ
Q7. Find the HCF of 225 and 375.
A. 25
B. 50
C. 75
D. 125
Correct Answer: C. 75
225=75×3 and 375=75×5. HCF=75. Practice MCQ
Q8. Find the LCM of 32 and 48.
A. 64
B. 72
C. 96
D. 144
Correct Answer: C. 96
32=2⁵ and 48=2⁴×3. LCM=2⁵×3=96. Practice MCQ
Q9. Find the HCF of 128 and 160.
A. 16
B. 24
C. 32
D. 40
Correct Answer: C. 32
HCF(128,160)=32. Practice MCQ
Q10. Find the LCM of 14, 35 and 20.
A. 70
B. 105
C. 140
D. 280
Correct Answer: C. 140
LCM(14,35,20)=140. Practice MCQ
Q11. The numbers 11 and 16 are co-prime. What is their LCM?
A. 88
B. 132
C. 176
D. 192
Correct Answer: C. 176
For co-prime numbers, LCM equals their product: 11×16=176. Practice MCQ
Q12. What is the HCF of 24 and 25?
A. 1
B. 2
C. 5
D. 12
Correct Answer: A. 1
Consecutive positive integers are co-prime. Therefore HCF=1. Practice MCQ
Q13. Since 24 divides 168 exactly, what is HCF(24,168)?
A. 12
B. 24
C. 48
D. 168
Correct Answer: B. 24
When the smaller number divides the larger exactly, HCF equals the smaller number: 24. Practice MCQ
Q14. Since 24 divides 168 exactly, what is LCM(24,168)?
A. 24
B. 84
C. 168
D. 336
Correct Answer: C. 168
When one number divides the other, LCM equals the larger number: 168. Practice MCQ
Q15. The product of two numbers is 7560 and their HCF is 18. Find their LCM.
A. 360
B. 400
C. 420
D. 540
Correct Answer: C. 420
LCM=7560÷18=420. Practice MCQ
Q16. The HCF and LCM of two numbers are 12 and 180. If one number is 45, find the other.
A. 36
B. 42
C. 48
D. 60
Correct Answer: C. 48
Other number=(12×180)/45=48. Practice MCQ
Q17. If the HCF of two integers is 14, which of the following cannot be their LCM?
A. 84
B. 98
C. 126
D. 132
Correct Answer: D. 132
LCM must be divisible by HCF. 132 is not divisible by 14. Practice MCQ
Q18. Two co-prime numbers have product 391. Find their LCM.
A. 17
B. 23
C. 391
D. 782
Correct Answer: C. 391
For co-prime numbers, HCF=1, so LCM equals their product=391. Practice MCQ
Q19. What is the LCM of 64 and 64?
A. 1
B. 32
C. 64
D. 128
Correct Answer: C. 64
LCM(a,a)=a. Therefore LCM=64. Practice MCQ
Q20. What is HCF(1,999)?
A. 1
B. 9
C. 111
D. 999
Correct Answer: A. 1
HCF(1,n)=1 for every positive integer n. Questions 21–40: Prime factorisation, division methods, fractions, decimals and special forms.
Practice MCQ
Q21. If A=24×32 and B=23×3×5, what is HCF(A,B)?
A. 12
B. 24
C. 36
D. 72
Correct Answer: B. 24
HCF=2³×3=24. Practice MCQ
Q22. For the same A and B, what is their LCM?
A. 360
B. 540
C. 720
D. 1440
Correct Answer: C. 720
LCM=2⁴×3²×5=720. Practice MCQ
Q23. Find the HCF of 867 and 255 by successive division.
A. 17
B. 34
C. 51
D. 102
Correct Answer: C. 51
867=255×3+102; 255=102×2+51; 102=51×2. HCF=51. Practice MCQ
Q24. Find the HCF of 252 and 198.
A. 9
B. 18
C. 27
D. 36
Correct Answer: B. 18
Successive division gives the last non-zero divisor 18. Practice MCQ
Q25. Find the LCM of 12, 18 and 30.
A. 90
B. 120
C. 180
D. 360
Correct Answer: C. 180
LCM=2²×3²×5=180. Practice MCQ
Q26. Find the HCF of 24, 36 and 60.
A. 6
B. 12
C. 18
D. 24
Correct Answer: B. 12
HCF(24,36,60)=12. Practice MCQ
Q27. Find the HCF of 2/3, 4/9 and 10/21.
A. 1/63
B. 2/63
C. 2/21
D. 4/63
Correct Answer: B. 2/63
HCF numerators=2; LCM denominators=63. Answer=2/63. Practice MCQ
Q28. Find the LCM of 3/4, 5/6 and 7/8.
A. 35/2
B. 105/4
C. 105/2
D. 210
Correct Answer: C. 105/2
LCM numerators=105; HCF denominators=2. Therefore LCM=105/2. Practice MCQ
Q29. Find the HCF of 3/4, 5/6 and 7/8.
A. 1/12
B. 1/18
C. 1/24
D. 1/48
Correct Answer: C. 1/24
HCF numerators=1; LCM denominators=24. Hence 1/24. Practice MCQ
Q30. Find the LCM of 1/8 and 1/12.
A. 1/24
B. 1/12
C. 1/8
D. 1/4
Correct Answer: D. 1/4
LCM numerators=1 and HCF(8,12)=4. Therefore LCM=1/4. Practice MCQ
Q31. Find the HCF of 1.2, 1.8 and 3.0.
A. 0.3
B. 0.4
C. 0.6
D. 1.2
Correct Answer: C. 0.6
Scale by10: HCF(12,18,30)=6. Therefore required HCF=0.6. Practice MCQ
Q32. Find the LCM of 0.4, 0.6 and 1.2.
A. 0.6
B. 1.0
C. 1.2
D. 2.4
Correct Answer: C. 1.2
Scale by10: LCM(4,6,12)=12. Required LCM=1.2. Practice MCQ
Q33. Find the HCF of 0.75, 1.25 and 2.00.
A. 0.05
B. 0.25
C. 0.50
D. 0.75
Correct Answer: B. 0.25
Scale by100: HCF(75,125,200)=25. Therefore 0.25. Practice MCQ
Q34. Find the LCM of 0.25, 0.5 and 1.5.
A. 0.75
B. 1.0
C. 1.5
D. 3.0
Correct Answer: C. 1.5
Scale by100: LCM(25,50,150)=150. Therefore 1.5. Practice MCQ
Q35. Find the HCF of 1 1/2 and 2 1/4.
A. 1/2
B. 3/4
C. 3/2
D. 9/4
Correct Answer: B. 3/4
Convert to3/2 and9/4. HCF=3/4. Practice MCQ
Q36. Find the LCM of 1 1/2 and 2 1/4.
A. 3
B. 7/2
C. 4
D. 9/2
Correct Answer: D. 9/2
Using3/2 and9/4, LCM=9/2=4 1/2. Practice MCQ
Q37. Find the HCF of 0.5, 3/4 and 1.25.
A. 0.10
B. 0.20
C. 0.25
D. 0.50
Correct Answer: C. 0.25
3/4=0.75. HCF of0.50,0.75,1.25 is 0.25. Practice MCQ
Q38. Find the LCM of 2/3, 5/6 and 7/9.
A. 35/3
B. 70/3
C. 35/9
D. 70/9
Correct Answer: B. 70/3
LCM numerators=70; HCF denominators=3. Answer=70/3. Practice MCQ
Q39. Find the LCM of 1/9 and 1/15.
A. 1/45
B. 1/15
C. 1/9
D. 1/3
Correct Answer: D. 1/3
LCM=1/HCF(9,15)=1/3. Practice MCQ
Q40. Find the HCF of 1/8 and 1/12.
A. 1/48
B. 1/24
C. 1/12
D. 1/4
Correct Answer: B. 1/24
HCF=1/LCM(8,12)=1/24. Questions 21–40: Prime factorisation, division methods, fractions, decimals and special forms.
Practice MCQ
Q21. If A=24×32 and B=23×3×5, what is HCF(A,B)?
A. 12
B. 24
C. 36
D. 72
Correct Answer: B. 24
HCF=2³×3=24. Practice MCQ
Q22. For the same A and B, what is their LCM?
A. 360
B. 540
C. 720
D. 1440
Correct Answer: C. 720
LCM=2⁴×3²×5=720. Practice MCQ
Q23. Find the HCF of 867 and 255 by successive division.
A. 17
B. 34
C. 51
D. 102
Correct Answer: C. 51
867=255×3+102; 255=102×2+51; 102=51×2. HCF=51. Practice MCQ
Q24. Find the HCF of 252 and 198.
A. 9
B. 18
C. 27
D. 36
Correct Answer: B. 18
Successive division gives the last non-zero divisor 18. Practice MCQ
Q25. Find the LCM of 12, 18 and 30.
A. 90
B. 120
C. 180
D. 360
Correct Answer: C. 180
LCM=2²×3²×5=180. Practice MCQ
Q26. Find the HCF of 24, 36 and 60.
A. 6
B. 12
C. 18
D. 24
Correct Answer: B. 12
HCF(24,36,60)=12. Practice MCQ
Q27. Find the HCF of 2/3, 4/9 and 10/21.
A. 1/63
B. 2/63
C. 2/21
D. 4/63
Correct Answer: B. 2/63
HCF numerators=2; LCM denominators=63. Answer=2/63. Practice MCQ
Q28. Find the LCM of 3/4, 5/6 and 7/8.
A. 35/2
B. 105/4
C. 105/2
D. 210
Correct Answer: C. 105/2
LCM numerators=105; HCF denominators=2. Therefore LCM=105/2. Practice MCQ
Q29. Find the HCF of 3/4, 5/6 and 7/8.
A. 1/12
B. 1/18
C. 1/24
D. 1/48
Correct Answer: C. 1/24
HCF numerators=1; LCM denominators=24. Hence 1/24. Practice MCQ
Q30. Find the LCM of 1/8 and 1/12.
A. 1/24
B. 1/12
C. 1/8
D. 1/4
Correct Answer: D. 1/4
LCM numerators=1 and HCF(8,12)=4. Therefore LCM=1/4. Practice MCQ
Q31. Find the HCF of 1.2, 1.8 and 3.0.
A. 0.3
B. 0.4
C. 0.6
D. 1.2
Correct Answer: C. 0.6
Scale by10: HCF(12,18,30)=6. Therefore required HCF=0.6. Practice MCQ
Q32. Find the LCM of 0.4, 0.6 and 1.2.
A. 0.6
B. 1.0
C. 1.2
D. 2.4
Correct Answer: C. 1.2
Scale by10: LCM(4,6,12)=12. Required LCM=1.2. Practice MCQ
Q33. Find the HCF of 0.75, 1.25 and 2.00.
A. 0.05
B. 0.25
C. 0.50
D. 0.75
Correct Answer: B. 0.25
Scale by100: HCF(75,125,200)=25. Therefore 0.25. Practice MCQ
Q34. Find the LCM of 0.25, 0.5 and 1.5.
A. 0.75
B. 1.0
C. 1.5
D. 3.0
Correct Answer: C. 1.5
Scale by100: LCM(25,50,150)=150. Therefore 1.5. Practice MCQ
Q35. Find the HCF of 1 1/2 and 2 1/4.
A. 1/2
B. 3/4
C. 3/2
D. 9/4
Correct Answer: B. 3/4
Convert to3/2 and9/4. HCF=3/4. Practice MCQ
Q36. Find the LCM of 1 1/2 and 2 1/4.
A. 3
B. 7/2
C. 4
D. 9/2
Correct Answer: D. 9/2
Using3/2 and9/4, LCM=9/2=4 1/2. Practice MCQ
Q37. Find the HCF of 0.5, 3/4 and 1.25.
A. 0.10
B. 0.20
C. 0.25
D. 0.50
Correct Answer: C. 0.25
3/4=0.75. HCF of0.50,0.75,1.25 is 0.25. Practice MCQ
Q38. Find the LCM of 2/3, 5/6 and 7/9.
A. 35/3
B. 70/3
C. 35/9
D. 70/9
Correct Answer: B. 70/3
LCM numerators=70; HCF denominators=3. Answer=70/3. Practice MCQ
Q39. Find the LCM of 1/9 and 1/15.
A. 1/45
B. 1/15
C. 1/9
D. 1/3
Correct Answer: D. 1/3
LCM=1/HCF(9,15)=1/3. Practice MCQ
Q40. Find the HCF of 1/8 and 1/12.
A. 1/48
B. 1/24
C. 1/12
D. 1/4
Correct Answer: B. 1/24
HCF=1/LCM(8,12)=1/24. Questions 61–80: Ratios, possible number pairs, pair counting, sums, differences and advanced HCF-LCM relations.
Practice MCQ
Q61. Two numbers are in the ratio 3:5 and their HCF is 8. Find the larger number.
A. 24
B. 32
C. 40
D. 48
Correct Answer: C. 40
Numbers=8×3 and8×5=24 and40. Practice MCQ
Q62. For the same numbers, find their LCM.
A. 80
B. 96
C. 120
D. 240
Correct Answer: C. 120
LCM=8×3×5=120. Practice MCQ
Q63. Two numbers are in the ratio 4:7 and their LCM is 420. Find their HCF.
A. 10
B. 12
C. 15
D. 20
Correct Answer: C. 15
HCF=420/(4×7)=15. Practice MCQ
Q64. Two numbers are in the ratio 6:10 and their HCF is 14. Find their sum.
A. 84
B. 98
C. 112
D. 224
Correct Answer: C. 112
Reduce6:10 to3:5. Numbers=42 and70. Sum=112. Practice MCQ
Q65. The HCF and LCM of two numbers are 12 and 420. How many unordered pairs are possible?
A. 1
B. 2
C. 3
D. 4
Correct Answer: B. 2
420/12=35. Co-prime factor pairs are(1,35) and(5,7): 2 pairs. Practice MCQ
Q66. The HCF and LCM of two numbers are 18 and 1260. How many unordered pairs are possible?
A. 2
B. 3
C. 4
D. 8
Correct Answer: C. 4
1260/18=70=2×5×7. With3 distinct prime factors, pair count=2²=4. Practice MCQ
Q67. Two numbers have HCF 10, LCM 120 and difference 10. Find the larger number.
A. 30
B. 40
C. 50
D. 60
Correct Answer: B. 40
Reduced factors have product12 and difference1: 3 and4. Numbers=30 and40. Practice MCQ
Q68. Two numbers have HCF 15, LCM 180 and sum 105. Find their difference.
A. 10
B. 15
C. 20
D. 30
Correct Answer: B. 15
Reduced factors are3 and4. Numbers=45 and60. Difference=15. Practice MCQ
Q69. The product of two numbers is 6760 and their HCF is 13. How many unordered pairs are possible?
A. 1
B. 2
C. 3
D. 4
Correct Answer: B. 2
Reduced product=6760/13²=40. Co-prime pairs:(1,40),(5,8). Hence 2. Practice MCQ
Q70. Which pair has HCF 8 and LCM 360?
A. 24 and120
B. 40 and72
C. 48 and60
D. 16 and180
Correct Answer: B. 40 and 72
40=8×5 and72=8×9; 5 and9 are co-prime. LCM=8×5×9=360. Practice MCQ
Q71. The HCF and LCM of two positive numbers are 6 and 150. Find their mean proportional.
A. 20
B. 25
C. 30
D. 36
Correct Answer: C. 30
Mean proportional=√(6×150)=√900=30. Practice MCQ
Q72. Two numbers have sum 66, HCF 6 and LCM 180. Find the sum of their reciprocals.
A. 11/180
B. 11/90
C. 1/18
D. 1/30
Correct Answer: A. 11/180
Sum of reciprocals=66/(6×180)=66/1080=11/180. Practice MCQ
Q73. Among all pairs having HCF 12 and LCM 420, what is the minimum possible sum?
A. 96
B. 120
C. 144
D. 432
Correct Answer: C. 144
Valid pairs are(12,420) and(60,84). Minimum sum=60+84=144. Practice MCQ
Q74. For HCF 12 and LCM 420, what is the maximum possible difference between the two numbers?
A. 324
B. 360
C. 408
D. 420
Correct Answer: C. 408
The extreme valid pair is12 and420. Difference=408. Practice MCQ
Q75. For HCF 18 and LCM 900, what is the smaller number in the valid pair having the minimum sum?
A. 18
B. 36
C. 90
D. 180
Correct Answer: B. 36
900/18=50. Closest co-prime pair is2 and25, giving numbers 36 and450. Practice MCQ
Q76. If LCM/HCF=105, how many unordered number pairs are possible?
A. 2
B. 3
C. 4
D. 8
Correct Answer: C. 4
105=3×5×7 has3 distinct prime factors. Pair count=2²=4. Practice MCQ
Q77. Two numbers are in the ratio 8:12 and their HCF is 5. Find their LCM.
A. 20
B. 30
C. 40
D. 60
Correct Answer: B. 30
Reduce8:12 to2:3. Numbers=10 and15. LCM=30. Practice MCQ
Q78. Two numbers are in the ratio 9:15 and their HCF is 7. Find their product.
A. 525
B. 630
C. 735
D. 945
Correct Answer: C. 735
Reduce9:15 to3:5. Numbers=21 and35. Product=735. Practice MCQ
Q79. Two numbers are in the ratio 5:8 and their LCM is 280. Find their sum.
A. 63
B. 77
C. 91
D. 105
Correct Answer: C. 91
HCF=280/(5×8)=7. Numbers=35 and56. Sum=91. Practice MCQ
Q80. The HCF and LCM of two numbers are 9 and 540. Find the product of the numbers.
A. 3240
B. 4320
C. 4860
D. 5400
Correct Answer: C. 4860
Product=HCF×LCM=9×540=4860. Questions 81–100: Mixed revision, advanced divisibility, perfect-square/cube applications and final exam-level practice.
Practice MCQ
Q81. Find the HCF of 144, 216 and 360.
A. 36
B. 48
C. 72
D. 108
Correct Answer: C. 72
HCF(144,216,360)=72. Practice MCQ
Q82. Find the LCM of 16, 24 and 40.
A. 120
B. 160
C. 240
D. 480
Correct Answer: C. 240
LCM(16,24,40)=240. Practice MCQ
Q83. Find the HCF of 1001 and 1430.
A. 11
B. 13
C. 77
D. 143
Correct Answer: D. 143
1001=143×7 and1430=143×10. Therefore HCF=143. Practice MCQ
Q84. Find the LCM of 27, 45 and 60.
A. 270
B. 360
C. 540
D. 1080
Correct Answer: C. 540
LCM(27,45,60)=540. Practice MCQ
Q85. Find the HCF of 999 and 729.
A. 9
B. 18
C. 27
D. 81
Correct Answer: C. 27
HCF(999,729)=27. Practice MCQ
Q86. Find the LCM of 72, 90 and 120.
A. 180
B. 240
C. 360
D. 720
Correct Answer: C. 360
LCM(72,90,120)=360. Practice MCQ
Q87. The product of two numbers is 15120 and their HCF is 42. Find their LCM.
A. 240
B. 300
C. 360
D. 420
Correct Answer: C. 360
LCM=15120÷42=360. Practice MCQ
Q88. The HCF and LCM of two numbers are 24 and 840. If one number is 120, find the other.
A. 144
B. 156
C. 168
D. 192
Correct Answer: C. 168
Other number=(24×840)/120=168. Practice MCQ
Q89. Two co-prime numbers have product 437. What is their LCM?
A. 19
B. 23
C. 437
D. 874
Correct Answer: C. 437
For co-prime numbers LCM equals their product=437. Practice MCQ
Q90. Which pair has HCF 16 and LCM 240?
A. 32 and120
B. 48 and80
C. 64 and60
D. 40 and96
Correct Answer: B. 48 and 80
48=16×3 and80=16×5. Since3 and5 are co-prime, HCF=16 and LCM=240. Practice MCQ
Q91. What is the minimum possible sum of two numbers having HCF 18 and LCM 900?
A. 468
B. 486
C. 540
D. 918
Correct Answer: B. 486
Valid closest pair is36 and450. Minimum sum=486. Practice MCQ
Q92. What is the minimum possible difference between two numbers having HCF 12 and LCM 252?
A. 24
B. 36
C. 48
D. 84
Correct Answer: C. 48
252/12=21. Closest co-prime factors are3 and7, giving numbers36 and84. Difference=48. Practice MCQ
Q93. Find the least perfect-square number divisible by 12, 18 and 30.
A. 180
B. 360
C. 720
D. 900
Correct Answer: D. 900
LCM=180=2²×3²×5. Multiply by5 to obtain perfect square=900. Practice MCQ
Q94. Find the least perfect-cube number divisible by 8, 12 and 18.
A. 72
B. 108
C. 216
D. 432
Correct Answer: C. 216
LCM=72=2³×3². Multiply by3 to get2³×3³=216. Practice MCQ
Q95. Three signals repeat every 14 sec, 21 sec and 35 sec. When will they next occur together?
A. 105 sec
B. 140 sec
C. 210 sec
D. 420 sec
Correct Answer: C. 210 sec
LCM(14,21,35)=210 sec. Practice MCQ
Q96. Find the greatest number that divides 1002 and 1230 leaving remainder 6 in each case.
A. 6
B. 12
C. 18
D. 24
Correct Answer: B. 12
HCF(996,1224)=12. Practice MCQ
Q97. Find the least number greater than 30 that leaves remainder 5 when divided by 12, 18 and 30.
A. 95
B. 125
C. 185
D. 365
Correct Answer: C. 185
LCM(12,18,30)=180. Required number=180+5=185. Practice MCQ
Q98. Two numbers have HCF 15 and LCM 210. What is their minimum possible difference?
A. 45
B. 60
C. 75
D. 90
Correct Answer: C. 75
210/15=14. Closest co-prime pair is2,7, giving30 and105. Difference=75. Practice MCQ
Q99. The HCF and LCM of two numbers are 12 and 360. How many unordered pairs are possible?
A. 2
B. 3
C. 4
D. 6
Correct Answer: C. 4
360/12=30=2×3×5. Pair count=2³⁻¹=4. Practice MCQ
Q100. Two numbers are in the ratio 3:4 and their HCF is 6. What is the difference between their LCM and HCF?
A. 54
B. 60
C. 66
D. 72
Correct Answer: C. 66
Numbers=18,24; LCM=72 and HCF=6. Difference=72−6=66.
1. HCF एवं LCM एक नज़र में
HCF: दी गई सभी positive integers को exactly divide करने वाली greatest positive integer।
LCM: दी गई सभी positive integers से exactly divisible smallest positive integer।
HCF को GCD भी कहा जाता है।
2. Factor एवं Multiple
Factor: किसी number को exactly divide करता है।
Multiple: किसी number को integer से multiply करने पर प्राप्त होता है।
Exam Tip:
HCF का संबंध common factors से और LCM का संबंध common multiples से है।
3. Basic Size Rule
Positive integers a एवं b के लिए:
HCF(a,b) ≤ छोटी संख्या
LCM(a,b) ≥ बड़ी संख्या
4. जब एक Number दूसरे को Divide करे
यदि a, b को exactly divide करती है और a≤b है:
HCF(a,b)=a
LCM(a,b)=b
5. Co-prime Numbers
दो positive integers co-prime होते हैं यदि उनका HCF=1 हो।
उदाहरण:
8 एवं15 co-prime हैं। दोनों का prime होना आवश्यक नहीं।
6. Co-prime Numbers का LCM
यदि HCF(a,b)=1 है:
LCM(a,b)=ab
7. Consecutive Numbers
कोई भी दो consecutive positive integers co-prime होते हैं।
इसलिए:
HCF=1
LCM=उनका product
8. Equal Numbers एवं Number 1
HCF(a,a)=a
LCM(a,a)=a
HCF(1,n)=1
LCM(1,n)=n
9. Fundamental Product Relation
Exactly दो positive integers के लिए:
पहली संख्या × दूसरी संख्या = HCF × LCM
Exam Trap:
इस simple identity को तीन या अधिक numbers पर direct apply न करें।
10. Unknown Number ज्ञात करना
Unknown Number = (HCF × LCM) ÷ Known Number
11. HCF को LCM को Divide करना चाहिए
Valid positive integers के लिए LCM को HCF से exactly divisible होना चाहिए।
12. HCF का Standard Representation
यदि HCF=h है, तो numbers लिखें:
hm एवं hn,
जहाँ HCF(m,n)=1।
13. HCF Representation में LCM
यदि numbers=hm एवं hn और m,n co-prime हैं:
LCM=hmn
14. Common Scaling Property
HCF(ka,kb)=k×HCF(a,b)
LCM(ka,kb)=k×LCM(a,b)
15. Prime Factorisation से HCF
HCF = सभी numbers में common prime factors की lowest powers।
Memory Rule:
HCF → Common + Lowest
16. Prime Factorisation से LCM
LCM = सभी required prime factors की highest powers।
Memory Rule:
LCM → All + Highest
17. HCF का Ladder Method
HCF ladder method में केवल ऐसे factor से divide करें जो सभी current numbers को exactly divide करता हो।
सभी common divisors का product HCF होगा।
18. LCM का Ladder Method
LCM ladder method में prime factor एक या अधिक current numbers को divide कर सकता है।
बाकी numbers unchanged आगे जाते हैं।
19. Successive Division Method
दो numbers के लिए:
Larger ÷ Smaller → remainder
Previous divisor ÷ remainder → next remainder
Remainder 0 होने तक जारी रखें।
Last non-zero divisor = HCF।
20. Euclidean Division Identity
Dividend = Divisor × Quotient + Remainder
21. Reverse Successive Division
यदि successive quotients एवं final HCF दिए हों, तो numbers को backwards reconstruct करें:
Dividend = Divisor × Quotient + Remainder
Exam Tip:
Given quotients को केवल multiply न करें।
22. कई Numbers का Stepwise HCF
HCF(a,b,c)=HCF(HCF(a,b),c)
23. कई Numbers का Stepwise LCM
LCM(a,b,c)=LCM(LCM(a,b),c)
24. Fractions का HCF
Positive fractions के सामान्य competitive-exam convention में:
Fractions का HCF
= Numerators का HCF ÷ Denominators का LCM
Exam Tip:
जहाँ संभव हो fractions को पहले lowest terms में reduce करें।
25. Fractions का LCM
Fractions का LCM
= Numerators का LCM ÷ Denominators का HCF
Memory Rule:
Fractions का HCF → ऊपर HCF, नीचे LCM।
Fractions का LCM → ऊपर LCM, नीचे HCF।
26. Mixed Fractions
Mixed fractions को HCF या LCM निकालने से पहले improper fractions में convert करें।
Common Mistake:
Whole-number और fractional parts पर अलग-अलग calculation न करें।
27. Decimals का Common Scaling Rule
Terminating decimals के लिए:
1. Decimal places equal करें।
2. सभी values को same power of10 से multiply करें।
3. Integer HCF या LCM निकालें।
4. Answer को same scaling factor से divide करें।
28. Fractions एवं Decimals Together
Calculation से पहले सभी quantities को एक consistent exact form में convert करें।
Shortcut:
वह exact form चुनें जो calculation को सबसे सरल integers में बदल दे।
29. Reciprocal Relationship
Positive rational aptitude convention में:
Reciprocals का HCF = Original LCM का reciprocal
Reciprocals का LCM = Original HCF का reciprocal
30. Greatest Equal Measurement
Greatest possible equal measuring unit = Measurements का HCF।
31. Largest Equal Pieces
Identical pieces की maximum possible length या size = HCF।
32. Largest Square Tile
Same units में rectangular floor के लिए:
Largest square tile की side = HCF(length,breadth)
Tiles की संख्या:
(length ÷ tile side) × (breadth ÷ tile side)
33. Maximum Identical Groups
Maximum number of identical groups = Quantities का HCF।
Per Group Quantity:
हर original quantity को HCF से divide करें।
34. Repeated Events
Fixed intervals पर repeat होने वाले events के लिए:
सभी events फिर एक साथ होने का समय = Intervals का LCM।
35. Simultaneous Occurrences की Counting
यदि events time 0 पर together start हों और दोनों endpoints included हों:
Common occurrences
= floor(T ÷ LCM) + 1
Exam Trap:
+1 केवल तभी करें जब initial occurrence वास्तव में count हो।
36. Several Numbers से Divisible Least Number
कई positive integers से exactly divisible least positive number = उनका LCM।
37. Common Divisibility के लिए Least Addition
यदि L=LCM और N को L से divide करने पर remainder r मिले:
r≠0 होने पर:
Least addition = L−r
38. Common Divisibility के लिए Least Subtraction
यदि remainder=r है:
Least subtraction = r
39. Known Same Remainder — HCF
a,b,c को divide करने पर known same remainder r छोड़ने वाली greatest divisor:
HCF(a−r,b−r,c−r)
40. Unknown Same Remainder — HCF
यदि same remainder है लेकिन उसका value unknown है:
Pairwise differences का HCF लें।
41. Different Remainders — HCF
यदि a,b,c से क्रमशः r₁,r₂,r₃ remainders मिलें:
Required greatest divisor
=HCF(a−r₁,b−r₂,c−r₃)
Necessary Check:
Divisor सभी stated remainders से बड़ा होना चाहिए।
42. Same-Remainder LCM Form
यदि N को a,b,c से divide करने पर remainder r मिलता है:
N−r सभी divisors से divisible होगा।
इसलिए:
N=k×LCM(a,b,c)+r
Important:
Least value तय करने से पहले lower-bound एवं additional conditions check करें।
43. Ratio और HCF
यदि reduced ratio=m:n और HCF=h है:
Numbers=hm एवं hn
LCM=hmn
Exam Trap:
Ratio को पहले lowest terms में reduce करें।
44. Ratio और LCM
यदि reduced ratio=m:n और LCM=L है:
HCF=L/(mn)
45. HCF एवं LCM से Possible Number Pairs
यदि HCF=h और LCM=L है:
mn=L/h
L/h के factor pairs निकालें और केवल co-prime pairs रखें।
Original numbers=hm एवं hn।
46. Possible Pairs की Counting
N=LCM/HCF निकालें।
N के unordered co-prime factor pairs count करें।
Shortcut:
यदि N>1 में k distinct prime factors हों:
Unordered pairs=2k−1
47. Sum एवं Difference Conditions
Numbers=hm एवं hn हों तो:
Sum S → m+n=S/h
Difference D → |m−n|=D/h
48. HCF, LCM एवं Sum Together
यदि HCF=h, LCM=L एवं sum=S है:
mn=L/h
m+n=S/h
Useful Equation:
t2−(S/h)t+(L/h)=0
49. Derived Relationships
दो positive numbers a,b के लिए:
ab=HCF×LCM
Mean proportional=√(HCF×LCM)
यदि a+b=S है:
1/a+1/b
=S/(HCF×LCM)
50. Final 30-Second Revision Sheet
Core Formulas:
दो numbers का Product=HCF×LCM
Fractions का HCF=HCF(numerators)/LCM(denominators)
Fractions का LCM=LCM(numerators)/HCF(denominators)
Greatest equal measure → HCF
Largest square tile → HCF
Maximum identical groups → HCF
Events together again → LCM
Least divisible number → LCM
Known same remainder r → HCF(numbers−r)
Unknown same remainder → Pairwise differences का HCF
Different remainders → HCF(number−corresponding remainder)
यदि HCF=h:
Numbers=hm,hn जहाँ m,n co-prime
LCM=hmn
यदि HCF=h एवं LCM=L:
mn=L/h
Ratio m:n + HCF h → hm,hn
Ratio m:n + LCM L → HCF=L/(mn)
यदि L/H में k distinct prime factors हों:
Possible unordered pairs=2k−1
Final Exam Strategy:
पहले question का mathematical pattern पहचानें, फिर HCF या LCM चुनें, units एवं forms को consistent बनाएँ और उसके बाद calculation करें।
Verified Previous-Year Questions
Chapter 3 PYQ Bank
HCF-LCM की basic properties, prime factorisation, product relation, fractions, decimals, applications, remainder problems तथा advanced ratio-based questions शामिल हैं।
SSC GD PYQBasic HCF24 फरवरी 2024 · Shift I
प्रश्न 1. 84, 126 एवं168 का HCF ज्ञात करें।
A. 44
B. 49
C. 40
D. 42
सही उत्तर: D. 42
84=22×3×7, 126=2×32×7 और168=23×3×7।
Common primes की lowest powers लेने पर HCF=42।
SSC CGL PYQHCF-LCM Validity1 दिसंबर 2022 · Tier-I · Shift I
प्रश्न 2. दो numbers का HCF 12 है। निम्न में से कौन-सा उनका LCM नहीं हो सकता?
A. 72
B. 60
C. 90
D. 84
सही उत्तर: C. 90
LCM को HCF से exactly divisible होना चाहिए।
90, 12 से divisible नहीं है। इसलिए 90 LCM नहीं हो सकता।
SSC GD PYQHCF : LCM12 फरवरी 2019 · Shift II
प्रश्न 3. 16 एवं20 के HCF तथा LCM का ratio क्या है?
A. 1 : 25
B. 4 : 63
C. 1 : 20
D. 1 : 10
सही उत्तर: C. 1 : 20
HCF=4 और LCM=80।
Ratio=4:80=1:20।
SSC MTS PYQProduct Relation13 जुलाई 2022 · Shift I
प्रश्न 4. दो numbers का product 432 और HCF 6 है। उनका LCM ज्ञात करें।
A. 68
B. 48
C. 78
D. 72
सही उत्तर: D. 72
LCM=432÷6=72।
SSC MTS PYQProduct Relation4 मई 2023 · Shift II
प्रश्न 5. दो numbers का LCM 72 तथा product 1728 है। HCF ज्ञात करें।
A. 1
B. 18
C. 24
D. 12
सही उत्तर: C. 24
HCF=1728÷72=24।
SSC MTS PYQDerived Relation26 अक्टूबर 2021 · Shift I
प्रश्न 6. दो numbers का HCF 12 एवं LCM 48 है। Numbers के product का square root क्या होगा?
A. 16
B. 24
C. 48
D. 12
सही उत्तर: B. 24
Product=12×48=576।
√576=24।
SSC GD PYQHCF of Fractions13 जनवरी 2023 · Shift III
प्रश्न 7. 3/8, 5/12 एवं15/16 का HCF ज्ञात करें।
A. 1/12
B. 1/24
C. 1/48
D. 1/36
सही उत्तर: C. 1/48
Numerators का HCF=1।
Denominators का LCM=48।
Required HCF=1/48।
SSC MTS PYQLCM of Fractions9 अक्टूबर 2024 · Shift I
प्रश्न 8. 1/7, 2/5 एवं3/15 का LCM ज्ञात करें।
A. 6
B. 4
C. 5
D. 3
सही उत्तर: A. 6
Numerators का LCM=6 और denominators का HCF=1।
Required LCM=6।
RRB NTPC PYQLCM of Decimals15 जून 2026 · UG CBT-I · Shift III
प्रश्न 9. 5, 3.2 एवं0.03 का LCM ज्ञात करें।
A. 215
B. 185
C. 120
D. 240
सही उत्तर: D. 240
5.00,3.20 और0.03 को100 से multiply करें:
500,320,3।
इनका LCM=24000।
Required LCM=24000/100=240।
RRB NTPC PYQLCM of Decimals7 अगस्त 2025 · UG CBT-I · Shift I
प्रश्न 10. 15.4 एवं0.006 का LCM ज्ञात करें।
A. 0.462
B. 462
C. 46.2
D. 4.62
सही उत्तर: C. 46.2
15.4=77/5 तथा0.006=3/500।
LCM numerators=231 और HCF denominators=5।
LCM=231/5=46.2।
SSC GD PYQGreatest Measurement24 नवंबर 2021 · Shift I
प्रश्न 11. 4 m 90 cm, 3 m 85 cm एवं12 m 95 cm को exactly measure करने वाली greatest possible length क्या है?
A. 35 cm
B. 25 cm
C. 20 cm
D. 45 cm
सही उत्तर: A. 35 cm
Lengths=490,385,1295 cm।
इनका HCF=35 cm।
SSC GD PYQBells & Repeated Events31 जनवरी 2023 · Shift I
प्रश्न 12. तीन bells together start करती हैं तथा8 sec,10 sec एवं12 sec के intervals पर बजती हैं। Starting instant सहित30 minutes में कितनी बार together बजेंगी?
A. 14
B. 17
C. 16
D. 18
सही उत्तर: C. 16
LCM=120 sec।
30 minutes=1800 sec।
1800/120=15 intervals।
Initial ringing सहित=16।
SSC GD PYQSame Remainder26 नवंबर 2021 · Shift III
प्रश्न 13. 265,580 एवं825 को divide करने पर same remainder छोड़ने वाली greatest number ज्ञात करें।
A. 35
B. 55
C. 25
D. 45
सही उत्तर: A. 35
Differences=315,245,560।
इनका HCF=35।
SSC MTS PYQSame Remainder & LCM4 नवंबर 2024 · Shift II
प्रश्न 14. 20,15 एवं30 से divide करने पर प्रत्येक case में remainder6 छोड़ने वाली smallest number ज्ञात करें।
A. 36
B. 66
C. 56
D. 96
सही उत्तर: B. 66
LCM(20,15,30)=60।
Required number=60+6=66।
SSC GD PYQLeast Number3 मार्च 2019 · Shift III
प्रश्न 15. वह least positive number ज्ञात करें जिसमें8 add करने पर वह4,5,6 एवं7 से exactly divisible हो जाए।
A. 322
B. 312
C. 412
D. 422
सही उत्तर: C. 412
LCM(4,5,6,7)=420।
Required number=420−8=412।
SSC CGL PYQRatio & HCF1 दिसंबर 2022 · Tier-I · Shift III
प्रश्न 16. दो numbers का ratio6:5 तथा HCF3 है। उनका LCM ज्ञात करें।
A. 64
B. 110
C. 90
D. 80
सही उत्तर: C. 90
Numbers=18 और15।
LCM=3×6×5=90।
SSC CGL PYQRatio & HCF13 दिसंबर 2022 · Tier-I · Shift I
प्रश्न 17. दो numbers का HCF9 तथा ratio14:19 है। उनका LCM ज्ञात करें।
A. 2394
B. 3990
C. 1596
D. 3192
सही उत्तर: A. 2394
14 और19 co-prime हैं।
LCM=9×14×19=2394।
SSC MTS PYQHCF, LCM & Ratio20 जुलाई 2022 · Shift III
प्रश्न 18. दो numbers का HCF35 तथा LCM840 है। यदि उनका ratio3:8 है, तो smaller number ज्ञात करें।
A. 35
B. 280
C. 105
D. 210
सही उत्तर: C. 105
Numbers=35×3 और35×8
=105 एवं280।
Smaller number=105।
SSC MTS PYQHCF, LCM & Difference27 अक्टूबर 2021 · Shift I
प्रश्न 19. दो numbers का HCF6 एवं LCM90 है। यदि एक number दूसरी से12 अधिक है, तो larger number ज्ञात करें।
A. 12
B. 45
C. 30
D. 51
सही उत्तर: C. 30
Product=6×90=540।
Product540 और difference12 वाली numbers18 एवं30 हैं।
Larger number=30।
RRB Group D PYQAdvanced HCF-LCM1 सितंबर 2022 · Shift III
प्रश्न 20. दो numbers का ratio4:3 है तथा उनके HCF और LCM का product432 है। HCF एवं LCM के reciprocals का sum ज्ञात करें।
A. 13/72
B. 13/27
C. 13/2
D. 13/7
सही उत्तर: A. 13/72
Numbers=4x एवं3x मानें।
12x²=432 ⇒ x=6।
Numbers=24 एवं18।
HCF=6 तथा LCM=72।
1/6+1/72=13/72।
RRB Group D PYQReverse Successive Division6 सितंबर 2022 · Shift III
प्रश्न 21. Division method से दो numbers का HCF ज्ञात करते समय successive quotients 1, 5 एवं2 प्राप्त होते हैं तथा last divisor 15 है। दोनों numbers का LCM क्या है?
A. 2145
B. 2130
C. 3045
D. 2115
सही उत्तर: A. 2145
HCF=15 से reverse calculation करें।
15×2=30
30×5+15=165
165×1+30=195।
Numbers=165 एवं195।
LCM=(165×195)/15=2145।
RRB NTPC PYQUnknown Number18 जून 2026 · UG CBT-I · Shift III
प्रश्न 22. दो numbers का HCF 27 एवं LCM 324 है। यदि एक number 81 है, तो दूसरी number ज्ञात करें।
A. 120
B. 84
C. 108
D. 96
सही उत्तर: C. 108
दूसरी number=(27×324)/81=108।
RRB NTPC PYQProduct Relation7 अप्रैल 2021 · CBT-I · Shift I
प्रश्न 23. 248 एवं868 का LCM 1736 है। उनका HCF क्या है?
A. 124
B. 248
C. 1736
D. 868
सही उत्तर: A. 124
HCF=(248×868)/1736=124।
SSC GD PYQDifferent Remainders24 फरवरी 2024 · Shift III
प्रश्न 24. 60 एवं90 को divide करने पर क्रमशः remainder2 एवं3 छोड़ने वाली greatest number ज्ञात करें।
A. 17
B. 29
C. 19
D. 23
सही उत्तर: B. 29
60−2=58 तथा90−3=87।
HCF(58,87)=29।
SSC GD PYQDifferent Remainders7 फरवरी 2025 · Shift II
प्रश्न 25. 943 एवं1957 को divide करने पर क्रमशः remainders7 एवं1 छोड़ने वाली greatest number ज्ञात करें।
A. 12
B. 28
C. 10
D. 13
सही उत्तर: A. 12
943−7=936
1957−1=1956।
HCF(936,1956)=12।
SSC GD PYQSame Remainder11 फरवरी 2025 · Shift III
प्रश्न 26. 7516, 7635 एवं7992 को greatest number x से divide करने पर प्रत्येक case में same remainder प्राप्त होता है। x के digits का product क्या है?
A. 6
B. 4
C. 8
D. 9
सही उत्तर: D. 9
Differences:
119,357 एवं476।
इनका HCF=119, इसलिए x=119।
Digits का product=1×1×9=9।
SSC CGL PYQLCM & Perfect Square24 जुलाई 2023 · Tier-I · Shift II
प्रश्न 27. 2,3,5,6,9 एवं18 से divisible least perfect-square number ज्ञात करें।
A. 900
B. 400
C. 144
D. 3600
सही उत्तर: A. 900
Given numbers का LCM=90=2×3²×5।
Perfect square के लिए2 एवं5 की powers को even करना होगा।
90×2×5=900।
RRC Group D PYQDivisibility Special Case26 अक्टूबर 2018 · Shift III
प्रश्न 28. 26 एवं182 का LCM 182 है। उनका HCF क्या होगा?
A. 29
B. 13
C. 17
D. 26
सही उत्तर: D. 26
182=26×7।
26,182 को exactly divide करता है, इसलिए HCF=26।
SSC MTS PYQBells & Counting17 मई 2023 · Shift II
प्रश्न 29. पाँच bells together start करती हैं और5,10,15,20 एवं25 seconds के intervals पर बजती हैं। 1 hour 20 minutes में कितनी बार together बजेंगी?
A. 20 times
B. 15 times
C. 14 times
D. 17 times
सही उत्तर: D. 17 times
Intervals का LCM=300 sec=5 min।
Total time=80 min।
80÷5=16 intervals।
Initial toll सहित=17 times।
RRB NTPC PYQRepeated Events7 मई 2026 · UG CBT-I · Shift I
प्रश्न 30. दो bells क्रमशः1764 seconds एवं2268 seconds के intervals पर बजती हैं और शुरू में together बजती हैं। वे first time फिर together कब बजेंगी?
A. 4 घंटे 24 मिनट 36 सेकंड
B. 4 घंटे 18 मिनट 12 सेकंड
C. 4 घंटे 36 मिनट 24 सेकंड
D. 4 घंटे 12 मिनट 18 सेकंड
सही उत्तर: A. 4 घंटे 24 मिनट 36 सेकंड
1764=252×7 तथा2268=252×9।
LCM=15876 sec
=4 घंटे 24 मिनट 36 सेकंड।
RRB NTPC PYQLCM of Decimals4 सितंबर 2025 · UG CBT-I · Shift II
प्रश्न 31. 2.8 एवं0.217 का LCM ज्ञात करें।
A. 86.8
B. 8.68
C. 0.868
D. 868
सही उत्तर: A. 86.8
2.800 एवं0.217 को1000 से scale करें:
2800 एवं217।
इनका LCM=86800।
Required LCM=86800/1000=86.8।
RRB NTPC PYQLCM of Decimals29 अगस्त 2025 · UG CBT-I · Shift I
प्रश्न 32. 5.3 एवं0.4 का LCM ज्ञात करें।
A. 0.212
B. 212
C. 21.2
D. 2.12
सही उत्तर: C. 21.2
10 से multiply करने पर53 एवं4 मिलते हैं।
LCM=212।
Required LCM=212/10=21.2।
SSC MTS PYQRatio & HCF5 नवंबर 2024 · Shift III
प्रश्न 33. दो numbers का ratio3:4 तथा HCF4 है। उनका LCM ज्ञात करें।
A. 24
B. 28
C. 38
D. 48
सही उत्तर: D. 48
Numbers=12 एवं16।
LCM=4×3×4=48।
SSC GD PYQAdvanced HCF-LCM13 फरवरी 2025 · Shift I
प्रश्न 34. दो numbers का LCM उनके HCF का126 times है। HCF=7 है तथा numbers का difference HCF का5 times है। Numbers का sum ज्ञात करें।
A. 611
B. 218
C. 161
D. 116
सही उत्तर: C. 161
Numbers=7m एवं7n मानें।
mn=126 तथा |m−n|=5।
Required co-prime factor pair=14 एवं9।
Sum=7(14+9)=161।
RRB NTPC PYQAdvanced Ratio Relation9 जून 2025 · Graduate CBT-I · Shift III
प्रश्न 35. दो numbers का ratio3:16 है और उनके HCF तथा LCM का product432 है। HCF एवं LCM के reciprocals का sum ज्ञात करें।
A. 49/159
B. 49/144
C. 49/151
D. 49/145
सही उत्तर: B. 49/144
Numbers=3x एवं16x।
HCF=x तथा LCM=48x।
48x²=432 ⇒ x=3।
HCF=3 तथा LCM=144।
1/3+1/144=49/144।
SSC GD PYQDirect LCM11 फरवरी 2025 · Shift I
प्रश्न 36. 75,45,180 एवं110 का LCM ज्ञात करें।
A. 9900
B. 9946
C. 9856
D. 9817
सही उत्तर: A. 9900
LCM=2²×3²×5²×11=9900।
SSC GD PYQProduct Relation11 फरवरी 2025 · Shift II
प्रश्न 37. दो numbers का product5184 तथा HCF24 है। उनका LCM ज्ञात करें।
A. 216
B. 324
C. 125
D. 343
सही उत्तर: A. 216
LCM=5184/24=216।
SSC GD PYQDirect LCM20 फरवरी 2025 · Shift II
प्रश्न 38. 46,68,136 एवं138 का LCM ज्ञात करें।
A. 9455
B. 9329
C. 9406
D. 9384
सही उत्तर: D. 9384
46=2×23, 68=2²×17, 136=2³×17 तथा138=2×3×23।
LCM=2³×3×17×23=9384।
RRB Group D PYQPrime Factorisation9 फरवरी 2026 · Shift I
प्रश्न 39. 45,63 एवं81 का LCM ज्ञात करें।
A. 2545
B. 2455
C. 2385
D. 2835
सही उत्तर: D. 2835
45=3²×5, 63=3²×7 एवं81=3⁴।
LCM=3⁴×5×7=2835।
SSC GD PYQLeast Divisible Number22 फरवरी 2024 · Shift II
प्रश्न 40. 27,36,56,12 एवं10 से exactly divisible least number ज्ञात करें।
A. 7560
B. 8584
C. 5953
D. 7694
सही उत्तर: A. 7560
LCM=2³×3³×5×7=7560।
53. Practice MCQs
प्रश्न 1–20: Basic HCF-LCM concepts, direct calculations एवं fundamental properties.
Practice MCQप्रश्न 1. 72 एवं120 का HCF ज्ञात करें।
A. 12
B. 18
C. 24
D. 36
सही उत्तर: C. 24
HCF(72,120)=24। Practice MCQप्रश्न 2. 18 एवं24 का LCM ज्ञात करें।
A. 36
B. 48
C. 72
D. 96
सही उत्तर: C. 72
LCM=2³×3²=72। Practice MCQप्रश्न 3. 96 एवं144 का HCF ज्ञात करें।
A. 24
B. 36
C. 48
D. 72
Practice MCQप्रश्न 4. 12,15 एवं20 का LCM ज्ञात करें।
A. 30
B. 45
C. 60
D. 120
Practice MCQप्रश्न 5. 84 एवं126 का HCF ज्ञात करें।
A. 21
B. 28
C. 42
D. 63
सही उत्तर: C. 42
84=42×2 तथा126=42×3। HCF=42। Practice MCQप्रश्न 6. 21,28 एवं36 का LCM ज्ञात करें।
A. 126
B. 168
C. 252
D. 504
सही उत्तर: C. 252
LCM=252। Practice MCQप्रश्न 7. 225 एवं375 का HCF ज्ञात करें।
A. 25
B. 50
C. 75
D. 125
Practice MCQप्रश्न 8. 32 एवं48 का LCM ज्ञात करें।
A. 64
B. 72
C. 96
D. 144
Practice MCQप्रश्न 9. 128 एवं160 का HCF ज्ञात करें।
A. 16
B. 24
C. 32
D. 40
Practice MCQप्रश्न 10. 14,35 एवं20 का LCM ज्ञात करें।
A. 70
B. 105
C. 140
D. 280
सही उत्तर: C. 140
LCM=140। Practice MCQप्रश्न 11. 11 एवं16 co-prime हैं। उनका LCM क्या होगा?
A. 88
B. 132
C. 176
D. 192
सही उत्तर: C. 176
Co-prime numbers का LCM=product=11×16=176। Practice MCQप्रश्न 12. 24 एवं25 का HCF क्या है?
A. 1
B. 2
C. 5
D. 12
सही उत्तर: A. 1
Consecutive integers co-prime होते हैं। HCF=1। Practice MCQप्रश्न 13. 24,168 को exactly divide करता है। HCF(24,168) क्या होगा?
A. 12
B. 24
C. 48
D. 168
सही उत्तर: B. 24
Smaller number ही HCF है: 24। Practice MCQप्रश्न 14. 24 एवं168 का LCM क्या है?
A. 24
B. 84
C. 168
D. 336
सही उत्तर: C. 168
Larger number ही LCM है: 168। Practice MCQप्रश्न 15. दो numbers का product7560 एवं HCF18 है। उनका LCM ज्ञात करें।
A. 360
B. 400
C. 420
D. 540
सही उत्तर: C. 420
LCM=7560÷18=420। Practice MCQप्रश्न 16. HCF12, LCM180 तथा एक number45 है। दूसरी number ज्ञात करें।
A. 36
B. 42
C. 48
D. 60
सही उत्तर: C. 48
दूसरी number=(12×180)/45=48। Practice MCQप्रश्न 17. दो integers का HCF14 है। इनमें से कौन-सा LCM नहीं हो सकता?
A. 84
B. 98
C. 126
D. 132
सही उत्तर: D. 132
LCM को14 से divisible होना चाहिए;132 divisible नहीं है। Practice MCQप्रश्न 18. दो co-prime numbers का product391 है। उनका LCM क्या होगा?
A. 17
B. 23
C. 391
D. 782
सही उत्तर: C. 391
LCM=product=391। Practice MCQप्रश्न 19. 64 एवं64 का LCM क्या है?
A. 1
B. 32
C. 64
D. 128
सही उत्तर: C. 64
LCM(a,a)=a। इसलिए 64। Practice MCQप्रश्न 20. HCF(1,999) क्या है?
A. 1
B. 9
C. 111
D. 999
सही उत्तर: A. 1
HCF(1,n)=1। Practice MCQs
प्रश्न 1–20: Basic HCF-LCM concepts, direct calculations एवं fundamental properties.
Practice MCQप्रश्न 1. 72 एवं120 का HCF ज्ञात करें।
A. 12
B. 18
C. 24
D. 36
सही उत्तर: C. 24
HCF(72,120)=24। Practice MCQप्रश्न 2. 18 एवं24 का LCM ज्ञात करें।
A. 36
B. 48
C. 72
D. 96
सही उत्तर: C. 72
LCM=2³×3²=72। Practice MCQप्रश्न 3. 96 एवं144 का HCF ज्ञात करें।
A. 24
B. 36
C. 48
D. 72
Practice MCQप्रश्न 4. 12,15 एवं20 का LCM ज्ञात करें।
A. 30
B. 45
C. 60
D. 120
Practice MCQप्रश्न 5. 84 एवं126 का HCF ज्ञात करें।
A. 21
B. 28
C. 42
D. 63
सही उत्तर: C. 42
84=42×2 तथा126=42×3। HCF=42। Practice MCQप्रश्न 6. 21,28 एवं36 का LCM ज्ञात करें।
A. 126
B. 168
C. 252
D. 504
सही उत्तर: C. 252
LCM=252। Practice MCQप्रश्न 7. 225 एवं375 का HCF ज्ञात करें।
A. 25
B. 50
C. 75
D. 125
Practice MCQप्रश्न 8. 32 एवं48 का LCM ज्ञात करें।
A. 64
B. 72
C. 96
D. 144
Practice MCQप्रश्न 9. 128 एवं160 का HCF ज्ञात करें।
A. 16
B. 24
C. 32
D. 40
Practice MCQप्रश्न 10. 14,35 एवं20 का LCM ज्ञात करें।
A. 70
B. 105
C. 140
D. 280
सही उत्तर: C. 140
LCM=140। Practice MCQप्रश्न 11. 11 एवं16 co-prime हैं। उनका LCM क्या होगा?
A. 88
B. 132
C. 176
D. 192
सही उत्तर: C. 176
Co-prime numbers का LCM=product=11×16=176। Practice MCQप्रश्न 12. 24 एवं25 का HCF क्या है?
A. 1
B. 2
C. 5
D. 12
सही उत्तर: A. 1
Consecutive integers co-prime होते हैं। HCF=1। Practice MCQप्रश्न 13. 24,168 को exactly divide करता है। HCF(24,168) क्या होगा?
A. 12
B. 24
C. 48
D. 168
सही उत्तर: B. 24
Smaller number ही HCF है: 24। Practice MCQप्रश्न 14. 24 एवं168 का LCM क्या है?
A. 24
B. 84
C. 168
D. 336
सही उत्तर: C. 168
Larger number ही LCM है: 168। Practice MCQप्रश्न 15. दो numbers का product7560 एवं HCF18 है। उनका LCM ज्ञात करें।
A. 360
B. 400
C. 420
D. 540
सही उत्तर: C. 420
LCM=7560÷18=420। Practice MCQप्रश्न 16. HCF12, LCM180 तथा एक number45 है। दूसरी number ज्ञात करें।
A. 36
B. 42
C. 48
D. 60
सही उत्तर: C. 48
दूसरी number=(12×180)/45=48। Practice MCQप्रश्न 17. दो integers का HCF14 है। इनमें से कौन-सा LCM नहीं हो सकता?
A. 84
B. 98
C. 126
D. 132
सही उत्तर: D. 132
LCM को14 से divisible होना चाहिए;132 divisible नहीं है। Practice MCQप्रश्न 18. दो co-prime numbers का product391 है। उनका LCM क्या होगा?
A. 17
B. 23
C. 391
D. 782
सही उत्तर: C. 391
LCM=product=391। Practice MCQप्रश्न 19. 64 एवं64 का LCM क्या है?
A. 1
B. 32
C. 64
D. 128
Practice MCQप्रश्न 21. यदि A=24×32 तथा B=23×3×5 है, तो HCF(A,B) क्या है?
A. 12
B. 24
C. 36
D. 72
सही उत्तर: B. 24
HCF=2³×3=24। Practice MCQप्रश्न 22. इन्हीं A एवं B का LCM ज्ञात करें।
A. 360
B. 540
C. 720
D. 1440
सही उत्तर: C. 720
LCM=2⁴×3²×5=720। Practice MCQप्रश्न 23. Successive division द्वारा867 एवं255 का HCF ज्ञात करें।
A. 17
B. 34
C. 51
D. 102
सही उत्तर: C. 51
867=255×3+102;255=102×2+51। HCF=51। Practice MCQप्रश्न 24. 252 एवं198 का HCF ज्ञात करें।
A. 9
B. 18
C. 27
D. 36
Practice MCQप्रश्न 25. 12,18 एवं30 का LCM ज्ञात करें।
A. 90
B. 120
C. 180
D. 360
सही उत्तर: C. 180
LCM=180। Practice MCQप्रश्न 26. 24,36 एवं60 का HCF ज्ञात करें।
A. 6
B. 12
C. 18
D. 24
Practice MCQप्रश्न 27. 2/3,4/9 एवं10/21 का HCF ज्ञात करें।
A. 1/63
B. 2/63
C. 2/21
D. 4/63
सही उत्तर: B. 2/63
Numerators का HCF=2 एवं denominators का LCM=63। Answer=2/63। Practice MCQप्रश्न 28. 3/4,5/6 एवं7/8 का LCM ज्ञात करें।
A. 35/2
B. 105/4
C. 105/2
D. 210
सही उत्तर: C. 105/2
Numerators LCM=105 तथा denominators HCF=2। Answer=105/2। Practice MCQप्रश्न 29. 3/4,5/6 एवं7/8 का HCF ज्ञात करें।
A. 1/12
B. 1/18
C. 1/24
D. 1/48
सही उत्तर: C. 1/24
Required HCF=1/24। Practice MCQप्रश्न 30. 1/8 एवं1/12 का LCM ज्ञात करें।
A. 1/24
B. 1/12
C. 1/8
D. 1/4
सही उत्तर: D. 1/4
LCM=1/4। Practice MCQप्रश्न 31. 1.2,1.8 एवं3.0 का HCF ज्ञात करें।
A. 0.3
B. 0.4
C. 0.6
D. 1.2
सही उत्तर: C. 0.6
12,18,30 का HCF6 है। Required HCF=0.6। Practice MCQप्रश्न 32. 0.4,0.6 एवं1.2 का LCM ज्ञात करें।
A. 0.6
B. 1.0
C. 1.2
D. 2.4
सही उत्तर: C. 1.2
Required LCM=1.2। Practice MCQप्रश्न 33. 0.75,1.25 एवं2.00 का HCF ज्ञात करें।
A. 0.05
B. 0.25
C. 0.50
D. 0.75
सही उत्तर: B. 0.25
75,125,200 का HCF25 है। Answer=0.25। Practice MCQप्रश्न 34. 0.25,0.5 एवं1.5 का LCM ज्ञात करें।
A. 0.75
B. 1.0
C. 1.5
D. 3.0
सही उत्तर: C. 1.5
Required LCM=1.5। Practice MCQप्रश्न 35. 1 1/2 एवं2 1/4 का HCF ज्ञात करें।
A. 1/2
B. 3/4
C. 3/2
D. 9/4
सही उत्तर: B. 3/4
Fractions3/2 तथा9/4 हैं। HCF=3/4। Practice MCQप्रश्न 36. 1 1/2 एवं2 1/4 का LCM ज्ञात करें।
A. 3
B. 7/2
C. 4
D. 9/2
सही उत्तर: D. 9/2
LCM=9/2। Practice MCQप्रश्न 37. 0.5,3/4 एवं1.25 का HCF ज्ञात करें।
A. 0.10
B. 0.20
C. 0.25
D. 0.50
सही उत्तर: C. 0.25
3/4=0.75। Required HCF=0.25। Practice MCQप्रश्न 38. 2/3,5/6 एवं7/9 का LCM ज्ञात करें।
A. 35/3
B. 70/3
C. 35/9
D. 70/9
सही उत्तर: B. 70/3
Required LCM=70/3। Practice MCQप्रश्न 39. 1/9 एवं1/15 का LCM ज्ञात करें।
A. 1/45
B. 1/15
C. 1/9
D. 1/3
सही उत्तर: D. 1/3
LCM=1/3। Practice MCQप्रश्न 40. 1/8 एवं1/12 का HCF ज्ञात करें।
A. 1/48
B. 1/24
C. 1/12
D. 1/4
सही उत्तर: B. 1/24
Required HCF=1/24। प्रश्न 41–60: Greatest measurement, grouping, repeated events, divisibility adjustment एवं remainder applications.
Practice MCQप्रश्न 41. 84 m,126 m एवं210 m की ropes को maximum equal lengths में काटना है। प्रत्येक piece की length क्या होगी?
A. 21 m
B. 28 m
C. 42 m
D. 63 m
सही उत्तर: C. 42 m
HCF=42 m। Practice MCQप्रश्न 42. 8.4 m × 6.3 m floor को largest square tiles से cover करने के लिए कितनी tiles चाहिए?
A. 8
B. 10
C. 12
D. 16
सही उत्तर: C. 12
Tile side=210 cm। Number=4×3=12। Practice MCQप्रश्न 43. 48 red,72 blue एवं120 green balls से maximum identical packets बनाए जाएँ। कितने packets बनेंगे?
A. 12
B. 18
C. 24
D. 36
सही उत्तर: C. 24
Maximum packets=HCF=24। Practice MCQप्रश्न 44. Bells 12,18 एवं30 sec के intervals पर बजती हैं। वे next time together कब बजेंगी?
A. 90 sec
B. 120 sec
C. 150 sec
D. 180 sec
सही उत्तर: D. 180 sec
LCM=180 sec। Practice MCQप्रश्न 45. 8,10 एवं12 sec वाली bells together start करती हैं। Starting instant सहित30 minutes में कितनी बार together बजेंगी?
A. 15
B. 16
C. 17
D. 18
सही उत्तर: B. 16
15 intervals + initial occurrence=16। Practice MCQप्रश्न 46. 18,24 एवं30 से exactly divisible least positive number ज्ञात करें।
A. 180
B. 240
C. 360
D. 720
सही उत्तर: C. 360
LCM=360। Practice MCQप्रश्न 47. 758 में least कितना add करें ताकि result 8,12 एवं18 से divisible हो?
A. 28
B. 34
C. 38
D. 42
सही उत्तर: B. 34
LCM72 एवं remainder38। Addition=72−38=34। Practice MCQप्रश्न 48. 758 में से least कितना subtract करें ताकि result 8,12 एवं18 से divisible हो?
A. 34
B. 36
C. 38
D. 42
सही उत्तर: C. 38
Required subtraction=remainder=38। Practice MCQप्रश्न 49. 275,455 एवं635 को divide करने पर remainder5 छोड़ने वाली greatest number क्या है?
A. 45
B. 60
C. 90
D. 135
सही उत्तर: C. 90
HCF(270,450,630)=90। Practice MCQप्रश्न 50. 391,527 एवं731 को divide करने पर same remainder छोड़ने वाली greatest number क्या है?
A. 34
B. 51
C. 68
D. 102
सही उत्तर: C. 68
Differences136,204,340 का HCF=68। Practice MCQप्रश्न 51. 465,724 एवं983 को divide करने पर क्रमशः remainders3,4 एवं5 छोड़ने वाली greatest number क्या है?
A. 6
B. 8
C. 12
D. 18
सही उत्तर: A. 6
HCF(462,720,978)=6। Practice MCQप्रश्न 52. 18 से greater least number ज्ञात करें जो8,12 एवं18 से divide होने पर remainder3 छोड़े।
A. 72
B. 75
C. 147
D. 219
सही उत्तर: B. 75
LCM=72। Required number=72+3=75। Practice MCQप्रश्न 53. 5,10,15,20 एवं25 sec वाली bells together start करती हैं। 80 minutes में initial occurrence सहित कितनी बार together बजेंगी?
A. 15
B. 16
C. 17
D. 18
सही उत्तर: C. 17
Common interval5 min। 16 intervals + initial=17। Practice MCQप्रश्न 54. 7.2 m × 5.4 m floor को largest square tiles से cover करना है। कितनी tiles चाहिए?
A. 8
B. 10
C. 12
D. 16
सही उत्तर: C. 12
Largest side180 cm। Tiles=4×3=12। Practice MCQप्रश्न 55. 96 notebooks,144 pens एवं240 pencils को maximum identical sets में बाँटना है। कितने sets बनेंगे?
A. 24
B. 36
C. 48
D. 72
Practice MCQप्रश्न 56. Machines 24,36 एवं54 sec के intervals पर repeat करती हैं। वे again together कब चलेंगी?
A. 108 sec
B. 180 sec
C. 216 sec
D. 432 sec
सही उत्तर: C. 216 sec
LCM=216 sec। Practice MCQप्रश्न 57. 6,8 एवं15 से divisible least positive number ज्ञात करें।
A. 60
B. 90
C. 120
D. 240
सही उत्तर: C. 120
LCM=120। Practice MCQप्रश्न 58. 4.2 m,6.3 m एवं8.4 m को exactly measure करने वाली greatest length क्या है?
A. 1.4 m
B. 1.8 m
C. 2.1 m
D. 4.2 m
सही उत्तर: C. 2.1 m
HCF=210 cm=2.1 m। Practice MCQप्रश्न 59. 1780 एवं2452 को divide करने पर remainder4 छोड़ने वाली greatest number क्या है?
A. 24
B. 36
C. 48
D. 72
सही उत्तर: C. 48
HCF(1776,2448)=48। Practice MCQप्रश्न 60. 742,1102 एवं1462 को divide करने पर remainder2 छोड़ने वाली greatest number क्या है?
A. 10
B. 20
C. 30
D. 40
सही उत्तर: B. 20
HCF(740,1100,1460)=20। प्रश्न 61–80: Ratio, number pairs, pair counting, sum, difference एवं advanced HCF-LCM relations.
Practice MCQप्रश्न 61. दो numbers का ratio3:5 तथा HCF8 है। बड़ी number ज्ञात करें।
A. 24
B. 32
C. 40
D. 48
सही उत्तर: C. 40
Numbers=24 एवं40। Practice MCQप्रश्न 62. इन्हीं numbers का LCM ज्ञात करें।
A. 80
B. 96
C. 120
D. 240
सही उत्तर: C. 120
LCM=8×3×5=120। Practice MCQप्रश्न 63. Ratio4:7 तथा LCM420 है। HCF ज्ञात करें।
A. 10
B. 12
C. 15
D. 20
सही उत्तर: C. 15
HCF=420/(4×7)=15। Practice MCQप्रश्न 64. Ratio6:10 तथा HCF14 है। Numbers का sum ज्ञात करें।
A. 84
B. 98
C. 112
D. 224
सही उत्तर: C. 112
Ratio3:5। Numbers42 एवं70। Sum=112। Practice MCQप्रश्न 65. HCF12 एवं LCM420 हो तो कितने unordered pairs possible हैं?
A. 1
B. 2
C. 3
D. 4
सही उत्तर: B. 2
35 के co-prime factor pairs(1,35),(5,7)। इसलिए 2। Practice MCQप्रश्न 66. HCF18 एवं LCM1260 हो तो कितने unordered pairs possible हैं?
A. 2
B. 3
C. 4
D. 8
सही उत्तर: C. 4
70=2×5×7। Pair count=2²=4। Practice MCQप्रश्न 67. HCF10, LCM120 और difference10 है। बड़ी number ज्ञात करें।
A. 30
B. 40
C. 50
D. 60
सही उत्तर: B. 40
Numbers30 एवं40। Practice MCQप्रश्न 68. HCF15, LCM180 एवं sum105 है। Difference ज्ञात करें।
A. 10
B. 15
C. 20
D. 30
सही उत्तर: B. 15
Numbers45,60। Difference=15। Practice MCQप्रश्न 69. Product6760 एवं HCF13 है। कितने unordered pairs possible हैं?
A. 1
B. 2
C. 3
D. 4
सही उत्तर: B. 2
Reduced product40 के co-prime pairs(1,40),(5,8)। इसलिए 2। Practice MCQप्रश्न 70. किस pair का HCF8 एवं LCM360 है?
A. 24 एवं120
B. 40 एवं72
C. 48 एवं60
D. 16 एवं180
सही उत्तर: B. 40 एवं72
40=8×5 तथा72=8×9। Practice MCQप्रश्न 71. HCF6 एवं LCM150 है। Mean proportional ज्ञात करें।
A. 20
B. 25
C. 30
D. 36
सही उत्तर: C. 30
√(6×150)=30। Practice MCQप्रश्न 72. Sum66, HCF6 एवं LCM180 है। Reciprocals का sum ज्ञात करें।
A. 11/180
B. 11/90
C. 1/18
D. 1/30
सही उत्तर: A. 11/180
66/(6×180)=11/180। Practice MCQप्रश्न 73. HCF12 एवं LCM420 वाले pairs का minimum possible sum क्या है?
A. 96
B. 120
C. 144
D. 432
सही उत्तर: C. 144
Closest valid pair60,84। Sum=144। Practice MCQप्रश्न 74. HCF12 एवं LCM420 के लिए maximum possible difference क्या है?
A. 324
B. 360
C. 408
D. 420
सही उत्तर: C. 408
Extreme pair12 एवं420। Difference=408। Practice MCQप्रश्न 75. HCF18 और LCM900 वाले minimum-sum pair में smaller number क्या है?
A. 18
B. 36
C. 90
D. 180
सही उत्तर: B. 36
Required pair36 एवं450। Smaller number=36। Practice MCQप्रश्न 76. यदि LCM/HCF=105 है, तो कितने unordered pairs possible हैं?
A. 2
B. 3
C. 4
D. 8
सही उत्तर: C. 4
105=3×5×7। Pair count=4। Practice MCQप्रश्न 77. Ratio8:12 एवं HCF5 है। LCM ज्ञात करें।
A. 20
B. 30
C. 40
D. 60
सही उत्तर: B. 30
Reduced ratio2:3। Numbers10,15। LCM=30। Practice MCQप्रश्न 78. Ratio9:15 एवं HCF7 है। Numbers का product ज्ञात करें।
A. 525
B. 630
C. 735
D. 945
सही उत्तर: C. 735
Numbers21 एवं35। Product=735। Practice MCQप्रश्न 79. Ratio5:8 एवं LCM280 है। Numbers का sum ज्ञात करें।
A. 63
B. 77
C. 91
D. 105
सही उत्तर: C. 91
HCF7। Numbers35,56। Sum=91। Practice MCQप्रश्न 80. HCF9 एवं LCM540 है। Numbers का product ज्ञात करें।
A. 3240
B. 4320
C. 4860
D. 5400
सही उत्तर: C. 4860
Product=9×540=4860। प्रश्न 81–100: Mixed revision, advanced divisibility, perfect-square/cube applications एवं final exam-level practice.
Practice MCQप्रश्न 81. 144,216 एवं360 का HCF ज्ञात करें।
A. 36
B. 48
C. 72
D. 108
Practice MCQप्रश्न 82. 16,24 एवं40 का LCM ज्ञात करें।
A. 120
B. 160
C. 240
D. 480
सही उत्तर: C. 240
LCM=240। Practice MCQप्रश्न 83. 1001 एवं1430 का HCF ज्ञात करें।
A. 11
B. 13
C. 77
D. 143
सही उत्तर: D. 143
1001=143×7 तथा1430=143×10। HCF=143। Practice MCQप्रश्न 84. 27,45 एवं60 का LCM ज्ञात करें।
A. 270
B. 360
C. 540
D. 1080
सही उत्तर: C. 540
LCM=540। Practice MCQप्रश्न 85. 999 एवं729 का HCF ज्ञात करें।
A. 9
B. 18
C. 27
D. 81
Practice MCQप्रश्न 86. 72,90 एवं120 का LCM ज्ञात करें।
A. 180
B. 240
C. 360
D. 720
सही उत्तर: C. 360
LCM=360। Practice MCQप्रश्न 87. दो numbers का product15120 तथा HCF42 है। LCM ज्ञात करें।
A. 240
B. 300
C. 360
D. 420
सही उत्तर: C. 360
LCM=15120÷42=360। Practice MCQप्रश्न 88. HCF24, LCM840 और एक number120 है। दूसरी number ज्ञात करें।
A. 144
B. 156
C. 168
D. 192
सही उत्तर: C. 168
दूसरी number=(24×840)/120=168। Practice MCQप्रश्न 89. दो co-prime numbers का product437 है। उनका LCM क्या होगा?
A. 19
B. 23
C. 437
D. 874
सही उत्तर: C. 437
LCM=product=437। Practice MCQप्रश्न 90. किस pair का HCF16 एवं LCM240 है?
A. 32 एवं120
B. 48 एवं80
C. 64 एवं60
D. 40 एवं96
सही उत्तर: B. 48 एवं80
48=16×3 एवं80=16×5। HCF16, LCM240। Practice MCQप्रश्न 91. HCF18 एवं LCM900 वाले two numbers का minimum possible sum क्या है?
A. 468
B. 486
C. 540
D. 918
सही उत्तर: B. 486
Closest valid pair36 एवं450। Sum=486। Practice MCQप्रश्न 92. HCF12 एवं LCM252 वाले numbers का minimum possible difference क्या है?
A. 24
B. 36
C. 48
D. 84
सही उत्तर: C. 48
Pair36 एवं84। Difference=48। Practice MCQप्रश्न 93. 12,18 एवं30 से divisible least perfect-square number ज्ञात करें।
A. 180
B. 360
C. 720
D. 900
सही उत्तर: D. 900
LCM180। Perfect square बनाने के लिए×5 ⇒ 900। Practice MCQप्रश्न 94. 8,12 एवं18 से divisible least perfect-cube number ज्ञात करें।
A. 72
B. 108
C. 216
D. 432
सही उत्तर: C. 216
LCM72=2³×3²। ×3 करने पर 216। Practice MCQप्रश्न 95. तीन signals 14,21 एवं35 sec के intervals पर repeat करते हैं। वे next time together कब होंगे?
A. 105 sec
B. 140 sec
C. 210 sec
D. 420 sec
सही उत्तर: C. 210 sec
LCM=210 sec। Practice MCQप्रश्न 96. 1002 एवं1230 को divide करने पर remainder6 छोड़ने वाली greatest number क्या है?
A. 6
B. 12
C. 18
D. 24
सही उत्तर: B. 12
HCF(996,1224)=12। Practice MCQप्रश्न 97. 30 से greater least number ज्ञात करें जो12,18 एवं30 से divide होने पर remainder5 छोड़े।
A. 95
B. 125
C. 185
D. 365
सही उत्तर: C. 185
LCM180। Required number=180+5=185। Practice MCQप्रश्न 98. HCF15 एवं LCM210 वाले numbers का minimum possible difference क्या है?
A. 45
B. 60
C. 75
D. 90
सही उत्तर: C. 75
Closest valid pair30 एवं105। Difference=75। Practice MCQप्रश्न 99. HCF12 एवं LCM360 हो तो कितने unordered pairs possible हैं?
A. 2
B. 3
C. 4
D. 6
सही उत्तर: C. 4
LCM/HCF=30=2×3×5। Pair count=4। Practice MCQप्रश्न 100. दो numbers का ratio3:4 एवं HCF6 है। उनके LCM और HCF का difference क्या है?
A. 54
B. 60
C. 66
D. 72
सही उत्तर: C. 66
Numbers18 एवं24। LCM72 और HCF6। Difference=66।