1. Introduction to HCF and LCM
HCF and LCM are fundamental concepts of arithmetic used to study the relationship between factors and multiples of two or more positive integers. They form the basis of many competitive-exam questions involving divisibility, grouping, measurement, repeated events and number relationships.
Terminology:
HCF = Highest Common Factor
GCD = Greatest Common Divisor
HCF and GCD mean the same thing.
LCM = Least Common Multiple
Scope:
Unless stated otherwise, the HCF and LCM concepts in this chapter are discussed for positive integers.
2. Factors and Multiples — Basic Recap
A factor divides a number exactly, whereas a multiple is obtained by multiplying the number by an integer.
Example:
Factors of 12:
1, 2, 3, 4, 6, 12
Some multiples of 12:
12, 24, 36, 48, 60, ...
Key Difference:
Factors of a positive integer are finite in number.
Multiples of a positive integer continue indefinitely.
Exam Tip:
HCF is connected with common factors, whereas LCM is connected with common multiples.
3. Common Factors and Common Multiples
A common factor divides each of the given numbers exactly. A common multiple is divisible by each of the given numbers.
Example:
For 12 and 18:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
Example:
Some multiples of 6: 6, 12, 18, 24, 30, ...
Some multiples of 8: 8, 16, 24, 32, 40, ...
24 is the first positive common multiple.
4. Meaning of HCF
The Highest Common Factor of two or more positive integers is the greatest positive integer that divides all of them exactly.
Definition:
HCF = Greatest among all common factors.
Example:
Common factors of 18 and 24 are:
1, 2, 3, 6.
Therefore:
HCF(18,24)=6.
Interpretation:
HCF tells us the largest possible unit or common divisor that can divide all the given quantities exactly.
5. Meaning of LCM
The Least Common Multiple of two or more positive integers is the smallest positive integer that is exactly divisible by every given number.
Definition:
LCM = Smallest positive common multiple.
Example:
Multiples of 6:
6, 12, 18, 24, 30, ...
Multiples of 8:
8, 16, 24, 32, ...
Therefore:
LCM(6,8)=24.
Interpretation:
LCM tells us the earliest or smallest positive value at which different multiplication patterns can meet.
6. Size of HCF and LCM
The HCF and LCM of positive integers obey useful size restrictions.
For two positive integers a and b:
HCF(a,b) ≤ smaller of a and b
LCM(a,b) ≥ larger of a and b
Example:
For 12 and 18:
HCF=6, which is not greater than 12.
LCM=36, which is not smaller than 18.
Exam Trap:
If a proposed HCF is greater than the smaller number, or a proposed LCM is smaller than the larger number, the data are impossible.
7. When One Number Divides the Other
If one positive integer divides another exactly, the HCF and LCM can be identified immediately.
Important Property:
If a divides b exactly and a≤b, then:
HCF(a,b)=a
LCM(a,b)=b
Example:
12 divides 60 exactly.
Therefore:
HCF(12,60)=12
LCM(12,60)=60.
Shortcut:
When the smaller number completely divides the larger number, no lengthy HCF-LCM calculation is required.
8. Co-prime Numbers and HCF-LCM
Two positive integers are co-prime if their HCF is 1. The numbers themselves do not have to be prime.
Example:
8 and 15 are both composite/ordinary integers, but they have no common factor greater than 1.
Therefore:
HCF(8,15)=1.
Important Property:
If a and b are co-prime, then:
HCF(a,b)=1
LCM(a,b)=ab
Example:
HCF(8,15)=1.
Therefore:
LCM(8,15)=8×15=120.
Common Mistake:
Co-prime numbers need not both be prime numbers.
9. Important Special Cases
Same Numbers:
HCF(a,a)=a
LCM(a,a)=a
Example:
HCF(37,37)=37
LCM(37,37)=37
1 and Any Positive Integer n:
HCF(1,n)=1
LCM(1,n)=n
Consecutive Positive Integers:
Any two consecutive positive integers are co-prime.
Therefore their HCF is 1 and their LCM is their product.
Example:
14 and 15 are consecutive.
HCF=1
LCM=14×15=210.
Distinct Prime Numbers:
Two different prime numbers are co-prime.
Their HCF is 1 and their LCM is their product.
10. Fundamental Product Relation
For exactly two positive integers, an extremely important relation connects the two numbers with their HCF and LCM.
Fundamental Formula:
First Number × Second Number = HCF × LCM
Example:
For 18 and 24:
18×24=432.
HCF=6 and LCM=72.
6×72=432.
Thus:
18×24=HCF×LCM.
Useful Rearrangement:
Other Number = (HCF × LCM) ÷ Given Number
Very Important:
The simple formula
HCF × LCM = product of the numbers
is a direct identity for two numbers. Do not automatically apply the same formula to three or more numbers.
11. Numbers Written in HCF Form
If the HCF of two numbers is h, the two numbers can be represented as multiples of h.
Representation:
Numbers = hm and hn,
where m and n are co-prime.
Example:
Suppose the HCF of two numbers is 6.
If the numbers are 24 and 30:
24=6×4
30=6×5
The reduced numbers 4 and 5 are co-prime.
Exam Insight:
After dividing both numbers by their HCF, the remaining parts must be co-prime.
Note:
Advanced problems using this representation to construct possible number pairs will be studied later in Topic 3.5.
12. Multiplying Both Numbers by the Same Factor
When both numbers are multiplied by the same positive integer, their HCF and LCM are also multiplied by that integer.
Property:
HCF(ka,kb)=k×HCF(a,b)
LCM(ka,kb)=k×LCM(a,b)
Example:
HCF(6,10)=2 and LCM(6,10)=30.
Multiply both numbers by 4:
24 and 40.
HCF(24,40)=8=4×2
LCM(24,40)=120=4×30.
13. HCF Must Divide the LCM
For positive integers, the HCF is a divisor of every given number while the LCM is a multiple of every given number. Therefore the HCF also divides the LCM.
Necessary Condition:
LCM must be exactly divisible by HCF.
Example:
If HCF=12 and LCM=180,
180÷12=15.
So this basic condition is satisfied.
Exam Trap:
If a question claims HCF=18 and LCM=100 for a pair of integers, the data cannot be valid because 100 is not divisible by 18.
14. How to Recognise HCF and LCM Questions
One of the most useful exam skills is recognising whether a word problem naturally requires HCF or LCM.
Think HCF when the question asks for:
• Greatest possible equal size
• Largest measuring unit
• Maximum length of equal pieces
• Greatest number that divides given quantities exactly
• Largest equal grouping or distribution
Think LCM when the question asks for:
• Least number divisible by several numbers
• Earliest time repeated events occur together
• Bells, lights or machines repeating together
• Smallest common quantity
• Minimum value satisfying several divisibility conditions
Exam Tip:
The words “greatest” and “least” alone are not enough. Understand what is being maximised or minimised before selecting HCF or LCM.
Note:
These applications will be studied thoroughly in Topic 3.4.
15. Common Exam Traps
Trap 1:
Confusing a factor with a multiple.
Trap 2:
Thinking that co-prime numbers must both be prime.
Trap 3:
Using the product relation for three or more numbers as if it were the two-number formula.
Trap 4:
Accepting an HCF that does not divide both numbers.
Trap 5:
Accepting an LCM that is not divisible by each given number.
Trap 6:
Accepting an LCM that is not divisible by the HCF.
Trap 7:
Assuming HCF must always be smaller than both numbers. It can equal the smaller number when the smaller number divides the larger exactly.
Trap 8:
Assuming LCM must always be greater than both numbers. It can equal the larger number when one number divides the other.
16. Quick Revision
Remember:
• HCF = greatest common factor.
• GCD and HCF mean the same thing.
• LCM = smallest positive common multiple.
• HCF is based on common factors.
• LCM is based on common multiples.
• HCF cannot exceed the smaller positive integer.
• LCM cannot be smaller than the larger positive integer.
• If one number divides the other, HCF=smaller number and LCM=larger number.
• Co-prime numbers have HCF=1.
• LCM of two co-prime numbers equals their product.
• Consecutive integers are co-prime.
• Distinct prime numbers are co-prime.
• HCF(a,a)=LCM(a,a)=a.
• HCF(1,n)=1 and LCM(1,n)=n.
• For exactly two positive integers: Product of numbers = HCF×LCM.
• If HCF=h, numbers can be written as hm and hn where m,n are co-prime.
• HCF must divide LCM.
• Multiplying both numbers by k multiplies both HCF and LCM by k.
17. Verified Previous-Year Questions
SSC GD PYQ
Co-prime Property
25 January 2023 · Shift IV
Q1. The product of two co-prime numbers is 323. What is their LCM?
A. 161.5
B. 646
C. 323
D. 1
Correct Answer: C. 323
For two co-prime numbers:
HCF=1.
Since:
Product=HCF×LCM,
323=1×LCM.
Therefore:
LCM=323.
SSC CGL PYQ
HCF-LCM Product Relation
5 December 2022 · Tier-I · Shift IV
Q2. The LCM of two numbers is 4104 and their HCF is 9. If one number is 171, find the other number.
A. 218
B. 215
C. 220
D. 216
Correct Answer: D. 216
For two numbers:
First Number × Second Number = HCF × LCM.
Therefore:
Second Number
=(9×4104)/171
=216.
SSC CGL PYQ
HCF-LCM & Number Pair
21 April 2022 · Tier-I · Shift III
Q3. The LCM and HCF of two numbers are 90 and 15 respectively. If their sum is 75, find the greater number.
A. 75
B. 45
C. 60
D. 90
Correct Answer: B. 45
Product of the numbers:
15×90=1350.
The two numbers whose product is 1350 and sum is 75 are 30 and 45.
Therefore the greater number is 45.
SSC CGL PYQ
HCF-LCM Property
6 December 2022 · Tier-I · Shift IV
Q4. The sum of two numbers is 18 and their HCF and LCM are 3 and 54 respectively. What is the sum of their reciprocals?
A. 1/7
B. 1/11
C. 1/6
D. 1/9
Correct Answer: D. 1/9
Let the numbers be a and b.
ab=HCF×LCM
=3×54
=162.
Also:
a+b=18.
Therefore:
1/a+1/b
=(a+b)/ab
=18/162
=1/9.
18. Practice MCQs
Practice MCQ
Q1. What is the HCF of 18 and 24?
A. 2
B. 3
C. 6
D. 12
Correct Answer: C. 6
Common factors are 1, 2, 3 and 6. The greatest is 6.
Practice MCQ
Q2. What is the LCM of 6 and 8?
A. 12
B. 18
C. 24
D. 48
Correct Answer: C. 24
The smallest positive number divisible by both 6 and 8 is 24.
Practice MCQ
Q3. If 8 and 15 are co-prime, what is their LCM?
A. 60
B. 90
C. 120
D. 240
Correct Answer: C. 120
LCM of two co-prime numbers equals their product: 8×15=120.
Practice MCQ
Q4. If one number is 12 and the other is 60, what is their HCF?
A. 6
B. 10
C. 12
D. 60
Correct Answer: C. 12
12 divides 60 exactly. Therefore the smaller number itself is the HCF: 12.
Practice MCQ
Q5. What is the LCM of 37 and 37?
A. 1
B. 37
C. 74
D. 1369
Correct Answer: B. 37
For equal positive numbers, the HCF and LCM both equal the number itself. Therefore LCM=37.
Practice MCQ
Q6. The HCF and LCM of two numbers are 6 and 180. If one number is 30, find the other.
A. 24
B. 30
C. 36
D. 42
Correct Answer: C. 36
Other number=(6×180)/30=36.
Practice MCQ
Q7. What is the HCF of any two consecutive positive integers?
A. 0
B. 1
C. 2
D. Depends on the numbers
Correct Answer: B. 1
Two consecutive positive integers are always co-prime. Hence their HCF is 1.
Practice MCQ
Q8. If the HCF of two numbers is 12 and the numbers are written as 12m and 12n, then what must be true about m and n?
A. Both must be even
B. Both must be prime
C. They must be co-prime
D. They must be equal
Correct Answer: C. They must be co-prime
After the complete HCF is removed, the remaining factors cannot have another common factor greater than 1.
Practice MCQ
Q9. Which relation is always true for exactly two positive integers a and b?
A. HCF+LCM=ab
B. HCF×LCM=ab
C. HCF−LCM=ab
D. HCF÷LCM=ab
Correct Answer: B. HCF×LCM=ab
For exactly two positive integers, product of the numbers = HCF×LCM.
Practice MCQ
Q10. Which pair can have HCF 8 and LCM 120?
A. 16 and 40
B. 24 and 40
C. 32 and 48
D. 40 and 80
Correct Answer: B. 24 and 40
HCF(24,40)=8 and LCM(24,40)=120. Therefore 24 and 40 satisfy both conditions.
1. HCF एवं LCM का परिचय
HCF एवं LCM arithmetic की महत्वपूर्ण अवधारणाएँ हैं, जिनका उपयोग दो या अधिक positive integers के factors और multiples के बीच संबंध समझने के लिए किया जाता है। Competitive examinations में divisibility, grouping, measurement, repeated events तथा number relationships से जुड़े अनेक प्रश्न इन्हीं concepts पर आधारित होते हैं।
महत्वपूर्ण शब्दावली:
HCF = Highest Common Factor = महत्तम समापवर्तक
GCD = Greatest Common Divisor
HCF और GCD का mathematical अर्थ समान है।
LCM = Least Common Multiple = लघुत्तम समापवर्त्य
Scope:
जब तक question में अलग से न कहा गया हो, इस chapter में HCF एवं LCM की चर्चा positive integers के संदर्भ में की गई है।
2. Factors एवं Multiples — Basic Recap
Factor वह संख्या है जो किसी संख्या को पूर्णतः divide करती है, जबकि multiple किसी संख्या को integer से multiply करने पर प्राप्त होता है।
उदाहरण:
12 के factors:
1, 2, 3, 4, 6, 12
12 के कुछ multiples:
12, 24, 36, 48, 60, ...
मुख्य अंतर:
किसी positive integer के factors सीमित संख्या में होते हैं।
उसके multiples अनंत होते हैं।
Exam Tip:
HCF का संबंध common factors से है, जबकि LCM का संबंध common multiples से है।
3. Common Factors एवं Common Multiples
Common factor वह factor है जो दी गई सभी संख्याओं को exactly divide करता है। Common multiple वह संख्या है जो दी गई सभी संख्याओं से exactly divisible होती है।
उदाहरण:
12 और18 के लिए:
12 के factors: 1, 2, 3, 4, 6, 12
18 के factors: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
उदाहरण:
6 के multiples: 6, 12, 18, 24, 30, ...
8 के multiples: 8, 16, 24, 32, 40, ...
इनका पहला positive common multiple 24 है।
4. HCF का अर्थ
दो या अधिक positive integers का HCF वह greatest positive integer है जो उन सभी को exactly divide करता है।
Definition:
HCF = सभी common factors में सबसे बड़ा factor।
उदाहरण:
18 एवं24 के common factors:
1, 2, 3, 6।
इसलिए:
HCF(18,24)=6।
Interpretation:
HCF यह बताता है कि दी गई quantities को exactly divide करने वाला सबसे बड़ा common unit या divisor कौन-सा हो सकता है।
5. LCM का अर्थ
दो या अधिक positive integers का LCM वह smallest positive integer है जो दी गई प्रत्येक संख्या से exactly divisible होता है।
Definition:
LCM = सबसे छोटा positive common multiple।
उदाहरण:
6 के multiples:
6, 12, 18, 24, 30, ...
8 के multiples:
8, 16, 24, 32, ...
इसलिए:
LCM(6,8)=24।
Interpretation:
LCM वह smallest positive value बताता है जहाँ अलग-अलग multiplication patterns एक साथ मिल सकते हैं।
6. HCF एवं LCM का Size
दो positive integers a और b के लिए:
HCF(a,b) ≤ दोनों में छोटी संख्या
LCM(a,b) ≥ दोनों में बड़ी संख्या
उदाहरण:
12 और18 के लिए:
HCF=6, जो12 से बड़ा नहीं है।
LCM=36, जो18 से छोटा नहीं है।
Exam Trap:
यदि proposed HCF छोटी संख्या से भी बड़ा हो या proposed LCM बड़ी संख्या से भी छोटा हो, तो data valid नहीं हो सकता।
7. जब एक संख्या दूसरी को पूर्णतः Divide करे
महत्वपूर्ण Property:
यदि a, b को exactly divide करती है और a≤b है, तो:
HCF(a,b)=a
LCM(a,b)=b
उदाहरण:
12, 60 को exactly divide करता है।
इसलिए:
HCF(12,60)=12
LCM(12,60)=60।
Shortcut:
यदि smaller number larger number को exactly divide करती हो, तो अलग से lengthy HCF-LCM calculation की आवश्यकता नहीं है।
8. Co-prime Numbers एवं HCF-LCM
दो positive integers co-prime कहलाते हैं यदि उनका HCF 1 हो। दोनों numbers का prime होना आवश्यक नहीं है।
उदाहरण:
8 और15 के बीच 1 के अतिरिक्त कोई common factor नहीं है।
इसलिए:
HCF(8,15)=1।
महत्वपूर्ण Property:
यदि a और b co-prime हैं, तो:
HCF(a,b)=1
LCM(a,b)=ab
उदाहरण:
8 और15 co-prime हैं।
इसलिए:
LCM=8×15=120।
Common Mistake:
Co-prime numbers का स्वयं prime होना आवश्यक नहीं है।
9. महत्वपूर्ण Special Cases
Same Numbers:
HCF(a,a)=a
LCM(a,a)=a
उदाहरण:
HCF(37,37)=37
LCM(37,37)=37
1 एवं कोई Positive Integer n:
HCF(1,n)=1
LCM(1,n)=n
Consecutive Positive Integers:
कोई भी दो consecutive positive integers co-prime होते हैं।
इसलिए उनका HCF=1 तथा LCM=उनका product।
उदाहरण:
14 एवं15 consecutive हैं।
HCF=1
LCM=14×15=210।
Distinct Prime Numbers:
दो अलग prime numbers co-prime होते हैं।
उनका HCF=1 तथा LCM=उनका product होता है।
10. Fundamental Product Relation
Exactly दो positive integers के लिए HCF एवं LCM का एक अत्यंत महत्वपूर्ण relation होता है।
Fundamental Formula:
पहली संख्या × दूसरी संख्या = HCF × LCM
उदाहरण:
18 और24 के लिए:
18×24=432।
HCF=6 तथा LCM=72।
6×72=432।
इसलिए:
18×24=HCF×LCM।
Useful Rearrangement:
दूसरी संख्या = (HCF × LCM) ÷ दी गई संख्या
बहुत महत्वपूर्ण:
HCF×LCM = numbers का product वाला simple formula सीधे दो numbers के लिए उपयोग किया जाता है। इसे तीन या अधिक numbers पर इसी रूप में automatically apply न करें।
11. HCF के रूप में Numbers लिखना
यदि दो numbers का HCF h है, तो दोनों numbers को h के multiples के रूप में लिखा जा सकता है।
Representation:
Numbers = hm तथा hn,
जहाँ m एवं n co-prime होंगे।
उदाहरण:
24 एवं30 का HCF=6।
24=6×4
30=6×5
Reduced numbers 4 एवं5 co-prime हैं।
Exam Insight:
दोनों numbers से complete HCF divide करने के बाद बचे हुए parts का HCF 1 होना चाहिए।
Note:
इस representation से possible number pairs निकालने वाले advanced questions Topic 3.5 में विस्तार से पढ़े जाएँगे।
12. दोनों Numbers को Same Factor से Multiply करना
Property:
HCF(ka,kb)=k×HCF(a,b)
LCM(ka,kb)=k×LCM(a,b)
उदाहरण:
HCF(6,10)=2 तथा LCM(6,10)=30।
दोनों numbers को4 से multiply करें:
24 एवं40।
HCF(24,40)=8=4×2
LCM(24,40)=120=4×30।
13. HCF को LCM को Divide करना चाहिए
HCF प्रत्येक given number का divisor है और LCM प्रत्येक given number का multiple है। इसलिए HCF, LCM को भी exactly divide करता है।
Necessary Condition:
LCM को HCF से exactly divisible होना चाहिए।
उदाहरण:
यदि HCF=12 और LCM=180 है,
तो 180÷12=15।
इसलिए basic condition satisfied है।
Exam Trap:
यदि किसी pair के लिए HCF=18 और LCM=100 दिया गया हो, तो data valid नहीं हो सकता क्योंकि100, 18 से exactly divisible नहीं है।
14. HCF या LCM कब Use करें?
HCF के बारे में सोचें जब question पूछे:
• Greatest possible equal size
• Largest measuring unit
• Maximum length of equal pieces
• दी गई quantities को divide करने वाली greatest number
• Largest equal grouping या distribution
LCM के बारे में सोचें जब question पूछे:
• कई numbers से divisible least number
• Repeated events का earliest common time
• Bells, lights या machines का फिर एक साथ होना
• Smallest common quantity
• कई divisibility conditions satisfy करने वाली minimum value
Exam Tip:
केवल “greatest” या “least” शब्द देखकर decision न लें। पहले समझें कि question किस quantity को maximise या minimise कर रहा है।
Note:
इन applications को Topic 3.4 में विस्तार से पढ़ा जाएगा।
15. Common Exam Traps
Trap 1:
Factor और multiple में confusion करना।
Trap 2:
यह मान लेना कि co-prime numbers दोनों prime होने चाहिए।
Trap 3:
दो-number product relation को तीन या अधिक numbers पर same way apply करना।
Trap 4:
ऐसा HCF स्वीकार करना जो दोनों numbers को divide ही न करे।
Trap 5:
ऐसा LCM स्वीकार करना जो प्रत्येक given number से divisible न हो।
Trap 6:
ऐसा LCM स्वीकार करना जो HCF से divisible न हो।
Trap 7:
यह मानना कि HCF हमेशा दोनों numbers से strictly छोटा होगा। यदि smaller number larger को divide करती है तो HCF smaller number के बराबर हो सकता है।
Trap 8:
यह मानना कि LCM हमेशा दोनों numbers से strictly बड़ा होगा। यदि एक number दूसरी को divide करती है तो LCM larger number के बराबर हो सकता है।
16. Quick Revision
एक नज़र में:
• HCF = greatest common factor।
• HCF और GCD का अर्थ समान है।
• LCM = smallest positive common multiple।
• HCF common factors पर आधारित है।
• LCM common multiples पर आधारित है।
• HCF smaller positive integer से बड़ा नहीं हो सकता।
• LCM larger positive integer से छोटा नहीं हो सकता।
• यदि smaller number larger को exactly divide करे, तो HCF=smaller और LCM=larger।
• Co-prime numbers का HCF=1।
• दो co-prime numbers का LCM=उनका product।
• Consecutive integers co-prime होते हैं।
• Distinct prime numbers co-prime होते हैं।
• HCF(a,a)=LCM(a,a)=a।
• HCF(1,n)=1 तथा LCM(1,n)=n।
• Exactly दो positive integers के लिए: numbers का product=HCF×LCM।
• यदि HCF=h है, तो numbers को hm एवं hn लिखा जा सकता है जहाँ m,n co-prime होंगे।
• HCF को LCM को exactly divide करना चाहिए।
• दोनों numbers को k से multiply करने पर HCF एवं LCM भी k से multiply होते हैं।
17. Verified Previous-Year Questions
SSC GD PYQ
Co-prime Property
25 जनवरी 2023 · Shift IV
प्रश्न 1. दो co-prime numbers का product 323 है। उनका LCM क्या होगा?
A. 161.5
B. 646
C. 323
D. 1
सही उत्तर: C. 323
Co-prime numbers के लिए HCF=1।
Product=HCF×LCM।
323=1×LCM।
अतः:
LCM=323।
SSC CGL PYQ
HCF-LCM Product Relation
5 दिसंबर 2022 · Tier-I · Shift IV
प्रश्न 2. दो numbers का LCM 4104 तथा HCF 9 है। यदि एक number 171 है, तो दूसरी number ज्ञात करें।
A. 218
B. 215
C. 220
D. 216
सही उत्तर: D. 216
दो numbers के लिए:
First Number × Second Number = HCF × LCM।
इसलिए:
दूसरी number
=(9×4104)/171
=216।
SSC CGL PYQ
HCF-LCM & Number Pair
21 अप्रैल 2022 · Tier-I · Shift III
प्रश्न 3. दो numbers का LCM एवं HCF क्रमशः 90 और15 है। यदि उनका sum 75 है, तो बड़ी number ज्ञात करें।
A. 75
B. 45
C. 60
D. 90
सही उत्तर: B. 45
Numbers का product:
15×90=1350।
ऐसी दो numbers जिनका product1350 तथा sum75 है:
30 एवं45।
इसलिए बड़ी number=45।
SSC CGL PYQ
HCF-LCM Property
6 दिसंबर 2022 · Tier-I · Shift IV
प्रश्न 4. दो numbers का sum 18 है तथा उनका HCF एवं LCM क्रमशः 3 और54 है। उनके reciprocals का sum क्या होगा?
A. 1/7
B. 1/11
C. 1/6
D. 1/9
सही उत्तर: D. 1/9
Numbers को a एवं b मानें।
ab=HCF×LCM
=3×54
=162।
a+b=18।
अतः:
1/a+1/b
=(a+b)/ab
=18/162
=1/9।
18. Practice MCQs
Practice MCQ
प्रश्न 1. 18 एवं24 का HCF क्या है?
A. 2
B. 3
C. 6
D. 12
सही उत्तर: C. 6
Common factors 1,2,3 और6 हैं। इनमें greatest factor=6।
Practice MCQ
प्रश्न 2. 6 एवं8 का LCM क्या है?
A. 12
B. 18
C. 24
D. 48
सही उत्तर: C. 24
6 एवं8 दोनों से divisible smallest positive number=24।
Practice MCQ
प्रश्न 3. यदि 8 एवं15 co-prime हैं, तो उनका LCM क्या होगा?
A. 60
B. 90
C. 120
D. 240
सही उत्तर: C. 120
Co-prime numbers का LCM उनके product के बराबर होता है। 8×15=120।
Practice MCQ
प्रश्न 4. 12 एवं60 का HCF क्या है?
A. 6
B. 10
C. 12
D. 60
सही उत्तर: C. 12
12, 60 को exactly divide करता है। इसलिए HCF=12।
Practice MCQ
प्रश्न 5. 37 एवं37 का LCM क्या है?
A. 1
B. 37
C. 74
D. 1369
सही उत्तर: B. 37
Equal positive numbers का HCF एवं LCM दोनों उसी number के बराबर होते हैं। इसलिए LCM=37।
Practice MCQ
प्रश्न 6. दो numbers का HCF=6 तथा LCM=180 है। यदि एक number=30 है, तो दूसरी number ज्ञात करें।
A. 24
B. 30
C. 36
D. 42
सही उत्तर: C. 36
दूसरी number=(6×180)/30=36।
Practice MCQ
प्रश्न 7. किसी भी दो consecutive positive integers का HCF क्या होता है?
A. 0
B. 1
C. 2
D. Numbers पर निर्भर करता है
सही उत्तर: B. 1
दो consecutive positive integers हमेशा co-prime होते हैं। इसलिए उनका HCF=1।
Practice MCQ
प्रश्न 8. यदि दो numbers का HCF 12 है और numbers को 12m एवं12n लिखा गया है, तो m एवं n के बारे में क्या सही है?
A. दोनों even होंगे
B. दोनों prime होंगे
C. दोनों co-prime होंगे
D. दोनों equal होंगे
सही उत्तर: C. दोनों co-prime होंगे
Complete HCF remove करने के बाद remaining parts में कोई common factor greater than1 नहीं होना चाहिए।
Practice MCQ
प्रश्न 9. Exactly दो positive integers a एवं b के लिए कौन-सा relation हमेशा सही है?
A. HCF+LCM=ab
B. HCF×LCM=ab
C. HCF−LCM=ab
D. HCF÷LCM=ab
सही उत्तर: B. HCF×LCM=ab
Exactly दो positive integers के लिए numbers का product=HCF×LCM।
Practice MCQ
प्रश्न 10. निम्न में से किस pair का HCF 8 एवं LCM 120 है?
A. 16 एवं40
B. 24 एवं40
C. 32 एवं48
D. 40 एवं80
सही उत्तर: B. 24 एवं40
HCF(24,40)=8 तथा LCM(24,40)=120। इसलिए correct pair=24 एवं40।