Greatest/Smallest Number Problems & Important Number Properties
Questions involving the greatest or smallest possible number often depend on place value, digit arrangement, divisibility or remainder conditions. This topic also collects several important number properties that are repeatedly useful in JSSC, SSC, Railway and other competitive examinations.
1. Smallest Positive Integer
The smallest positive integer is:
1
There is no greatest positive integer because integers continue indefinitely.
2. Greatest and Smallest One-Digit Positive Numbers
The smallest positive one-digit number is:
1
The greatest one-digit number is:
9
3. Smallest n-Digit Positive Number
The smallest positive integer containing exactly n digits is:
10n−1
Examples:
- Smallest 2-digit number = 10
- Smallest 3-digit number = 100
- Smallest 5-digit number = 10,000
4. Greatest n-Digit Number
The greatest n-digit positive integer is:
10n − 1
Examples:
- Greatest 2-digit number = 99
- Greatest 3-digit number = 999
- Greatest 5-digit number = 99,999
5. Difference Between Greatest and Smallest n-Digit Numbers
The difference is:
(10n − 1) − 10n−1
= 9 × 10n−1 − 1
Example: For three-digit numbers:
999 − 100 = 899
6. Number Formation from Given Digits
To form the greatest number from given digits, generally arrange the digits in descending order.
Example: From 3, 8, 1 and 6:
Greatest number = 8631
7. Smallest Number from Given Non-Zero Digits
If none of the given digits is zero, arrange them in ascending order.
Example: From 7, 2, 9 and 4:
Smallest number = 2479
8. Special Rule When Zero is Present
A multi-digit number cannot begin with zero.
Therefore, to form the smallest number:
- Place the smallest non-zero digit first.
- Place all zeros immediately after it.
- Arrange the remaining digits in ascending order.
9. Example with Zero
Form the smallest number using 0, 5, 2, 8 and 0.
The smallest non-zero digit is 2.
Place 2 first, then both zeros, then 5 and 8:
20,058
Common Mistake: 00258 is not a five-digit number. Leading zeros do not create additional digits.
10. Greatest Number with Distinct Digits
To form the greatest number with distinct decimal digits, select the largest available digits and arrange them in descending order.
Example: Greatest five-digit number with no repeated digit:
98,765
11. Smallest Number with Distinct Digits
For a multi-digit number with distinct digits, zero should normally be placed immediately after the smallest possible non-zero leading digit.
Example: Smallest five-digit number with distinct digits:
10,234
12. Greatest and Smallest 10-Digit Numbers Using Every Digit Once
Using digits 0 to 9 exactly once:
Greatest = 9,876,543,210
Smallest = 1,023,456,789
13. When Repetition of Digits is Allowed
If repetition is allowed and any decimal digit may be used:
- Greatest n-digit number = 99...9
- Smallest n-digit positive number = 10...0
Thus:
Greatest = 10n − 1
Smallest = 10n−1
14. Greatest Number from an Allowed Set of Digits
If repetition is allowed, repeat the greatest allowed digit.
Example: Using only digits 2, 5 and 7, repetition allowed, greatest four-digit number:
7777
15. Smallest Number from an Allowed Set When Zero is Absent
If repetition is allowed and zero is unavailable, repeat the smallest allowed digit.
Example: Using digits 3, 6 and 8:
Smallest four-digit number = 3333
16. Smallest Number from an Allowed Set When Zero is Available
If zero and at least one non-zero digit are allowed, use the smallest non-zero digit first and zeros afterwards.
Example: Using digits 0, 4 and 9 with repetition allowed:
Smallest five-digit number = 40,000
17. Decimal Representation of a Two-Digit Number
If a is the tens digit and b is the units digit, the number is:
10a + b
where:
1 ≤ a ≤ 9, 0 ≤ b ≤ 9
18. Reversal of a Two-Digit Number
The reverse of:
10a + b
is:
10b + a
19. Difference Between a Two-Digit Number and its Reverse
Difference:
(10a + b) − (10b + a)
= 9(a − b)
Important Property: The difference between a two-digit number and its reversed number is always divisible by 9.
20. Sum of a Two-Digit Number and its Reverse
Sum:
(10a + b) + (10b + a)
= 11(a + b)
Important Property: The sum of a two-digit number and its reverse is always divisible by 11.
21. Example: Two-Digit Reversal
Consider 73 and 37.
Difference:
73 − 37 = 36 = 9 × 4
Sum:
73 + 37 = 110 = 11 × 10
22. Decimal Representation of a Three-Digit Number
If digits are a, b and c:
100a + 10b + c
Its reverse is:
100c + 10b + a
23. Difference Between a Three-Digit Number and its Reverse
Subtract:
(100a + 10b + c) − (100c + 10b + a)
= 99(a − c)
Important Property: The difference between a three-digit number and its reverse is always divisible by 99.
24. Appending a Digit to a Number
If digit d is appended to the right of an integer N:
New number = 10N + d
Example: Append 7 to 245:
10 × 245 + 7 = 2457
25. Appending Zero
Appending one zero multiplies a number by 10:
N → 10N
Appending k zeros gives:
N × 10k
26. Prefixing a Digit
If N is a k-digit number and digit d is placed before it, the new number is:
d × 10k + N
Example: Place 7 before 253:
7 × 1000 + 253 = 7253
27. Number Consisting Entirely of 1s
The n-digit number:
111...111
is called a repunit and equals:
(10n − 1)/9
Example:
1111 = (104 − 1)/9
28. Number Made by Repeating the Same Digit
An n-digit number consisting entirely of digit d is:
d(10n − 1)/9
Example:
7777 = 7(104 − 1)/9
29. Repeating a Two-Digit Block
If a two-digit number N is written twice consecutively, the resulting four-digit number is:
100N + N
= 101N
Example:
3535 = 35 × 101
30. Repeating a Three-Digit Block
If a three-digit number N is repeated twice:
1000N + N = 1001N
Since:
1001 = 7 × 11 × 13
Important Property: Every six-digit number of the form abcabc is divisible by 7, 11 and 13.
31. Example of Repeated Three-Digit Block
Consider:
357357
Then:
357357 = 357 × 1001
Therefore it is divisible by 7, 11 and 13.
32. Greatest n-Digit Multiple of a Number
Let:
G = 10n − 1
be the greatest n-digit number.
If G leaves remainder r when divided by d, then:
Greatest n-digit multiple of d = G − r
33. Example: Greatest Four-Digit Multiple of 37
Greatest four-digit number:
9999
Now:
9999 = 37 × 270 + 9
Therefore:
Greatest four-digit multiple = 9999 − 9 = 9990
34. Smallest n-Digit Multiple of a Number
Let:
S = 10n−1
be the smallest n-digit number.
If S is already divisible by d, then S itself is the answer.
If S leaves remainder r, where r ≠ 0, then:
Smallest n-digit multiple of d = S + (d − r)
35. Example: Smallest Four-Digit Multiple of 37
Smallest four-digit number:
1000
Since:
1000 = 37 × 27 + 1
we need:
37 − 1 = 36
more.
Therefore:
1000 + 36 = 1036
36. Greatest Number Not Exceeding N that is Divisible by d
If N leaves remainder r when divided by d:
Greatest multiple of d ≤ N = N − r
Example:
347 divided by 12 leaves remainder 11.
Therefore:
Greatest multiple of 12 not exceeding 347 = 347 − 11 = 336
37. Smallest Multiple of d Not Less than N
If N leaves remainder r:
- If r = 0, answer = N.
- If r ≠ 0, add d − r.
Thus:
Next multiple = N + (d − r)
38. Example: Smallest Multiple of 17 Greater than or Equal to 500
500 divided by 17 gives:
500 = 17 × 29 + 7
Required addition:
17 − 7 = 10
Therefore:
Required number = 510
39. Greatest n-Digit Number Leaving a Given Remainder
Suppose we need the greatest n-digit number that leaves remainder r when divided by d.
Let:
G = 10n − 1
Reduce G to the nearest number congruent to r modulo d.
Equivalently, subtract the remainder obtained when G−r is divided by d.
40. Example: Greatest Three-Digit Number Leaving Remainder 5 on Division by 17
Greatest three-digit number = 999.
We require:
N ≡ 5 (mod 17)
Now:
999 − 5 = 994
994 leaves remainder 8 when divided by 17.
Therefore:
N = 999 − 8 = 991
Check:
991 = 17 × 58 + 5
41. Smallest n-Digit Number Leaving a Given Remainder
Start with:
S = 10n−1
and increase it until its remainder upon division by d becomes the required remainder.
42. Example: Smallest Three-Digit Number Leaving Remainder 7 on Division by 13
Smallest three-digit number = 100.
Now:
100 = 13 × 7 + 9
We need remainder 7 rather than 9.
The next suitable residue occurs after adding:
7 − 9 ≡ 11 (mod 13)
Therefore:
100 + 11 = 111
Check:
111 = 13 × 8 + 7
43. Counting Multiples in a Range
The number of positive multiples of d not exceeding N is:
⌊N/d⌋
Therefore, the number of multiples of d from A to B inclusive is:
⌊B/d⌋ − ⌊(A−1)/d⌋
44. Example: Multiples of 7 from 100 to 500
Multiples of 7 up to 500:
⌊500/7⌋ = 71
Multiples up to 99:
⌊99/7⌋ = 14
Therefore:
71 − 14 = 57
45. Consecutive Integers
Consecutive integers can be represented as:
n, n+1, n+2, ...
Two consecutive integers are:
n and n+1
46. Product of Two Consecutive Integers
One of two consecutive integers must be even.
Therefore:
n(n+1) is always divisible by 2.
47. Product of Three Consecutive Integers
Among three consecutive integers:
- At least one is divisible by 2.
- Exactly one is divisible by 3.
Therefore:
n(n+1)(n+2) is always divisible by 6.
48. Product of Four Consecutive Integers
The product of any four consecutive integers is divisible by:
4! = 24
49. General Consecutive-Integer Property
The product of any k consecutive integers is always divisible by:
k!
Important Property: Product of k consecutive integers is divisible by k!.
50. Sum of First n Natural Numbers
1 + 2 + 3 + ... + n = n(n+1)/2
Example:
1 + 2 + ... + 100 = 100 × 101 / 2 = 5050
51. Sum of First n Odd Numbers
1 + 3 + 5 + ... + (2n−1) = n2
Important Property: The sum of the first n odd numbers is always a perfect square.
52. Sum of First n Even Numbers
2 + 4 + 6 + ... + 2n = n(n+1)
53. Sum of Squares of First n Natural Numbers
12 + 22 + ... + n2 = n(n+1)(2n+1)/6
54. Sum of Cubes of First n Natural Numbers
13 + 23 + ... + n3 = [n(n+1)/2]2
Beautiful Property: The sum of the first n cubes equals the square of the sum of the first n natural numbers.
55. Difference of Consecutive Squares
(n+1)2 − n2
= n2 + 2n + 1 − n2
= 2n + 1
Therefore, the difference between consecutive perfect squares is always odd.
56. Every Integer is Even or Odd
Every integer can be written uniquely in one of the forms:
2k
or:
2k + 1
These represent even and odd integers respectively.
57. Three Residue Forms Modulo 3
Every integer is of exactly one of the forms:
3k, 3k+1, 3k+2
This representation is frequently useful in proofs and remainder-based questions.
58. Important Special Number Facts
- 0 is even.
- 0 is neither positive nor negative.
- 1 is neither prime nor composite.
- 2 is the smallest prime number.
- 2 is the only even prime number.
- 4 is the smallest composite number.
59. Sum and Product Parity Rules
| Operation | Result |
|---|
| Even + Even | Even |
| Odd + Odd | Even |
| Even + Odd | Odd |
| Even × Any Integer | Even |
| Odd × Odd | Odd |
60. Difference Between Odd and Even Numbers
- Even − Even = Even
- Odd − Odd = Even
- Even − Odd = Odd
- Odd − Even = Odd
61. Square Parity
If n is even, n2 is even.
If n is odd, n2 is odd.
Thus:
n and n2 always have the same parity.
62. An Odd Square Leaves Remainder 1 on Division by 8
Every odd integer can be represented as 2k+1.
A useful standard property is:
Odd square ≡ 1 (mod 8)
Examples:
- 32 = 9 → remainder 1 on division by 8
- 52 = 25 → remainder 1
- 72 = 49 → remainder 1
63. Exam Strategy for Greatest/Smallest Problems
- Identify the basic upper or lower boundary.
- Check whether digits may repeat.
- Handle zero carefully when forming the smallest number.
- If divisibility is involved, use the remainder from the boundary.
- For greatest valid number, generally move downward.
- For smallest valid number, generally move upward.
- For a specified remainder, work with the required residue rather than testing numbers one by one.
64. Common Exam Traps
Trap 1: Smallest n-digit number is 10n−1, not 0 followed by digits.
Trap 2: Greatest n-digit number is 10n−1.
Trap 3: When zero is present, do not place it at the beginning of the smallest multi-digit number.
Trap 4: For distinct digits, a digit cannot be reused unless repetition is explicitly allowed.
Trap 5: To find the greatest multiple below a number, subtract its remainder.
Trap 6: To find the next multiple above a non-multiple, add divisor − remainder.
Trap 7: A two-digit number is 10a+b, not a+b.
Trap 8: The difference between a two-digit number and its reverse is divisible by 9.
Trap 9: A number of the form abcabc is 1001 × abc and is divisible by 7, 11 and 13.
Trap 10: Product of k consecutive integers is always divisible by k!, but this does not mean it is always equal to k!.
65. Quick Revision
- Smallest n-digit positive number = 10n−1.
- Greatest n-digit number = 10n−1.
- To form the greatest number from given digits, arrange them in descending order.
- To form the smallest number without zero, arrange digits in ascending order.
- If zero is present, place the smallest non-zero digit first, then zeros.
- Greatest five-digit number with distinct digits = 98765.
- Smallest five-digit number with distinct digits = 10234.
- Two-digit number = 10a+b.
- Reverse = 10b+a.
- Difference of a two-digit number and its reverse is divisible by 9.
- Sum of a two-digit number and its reverse is divisible by 11.
- Three-digit number = 100a+10b+c.
- Difference from its reverse = 99(a−c).
- Appending digit d to N gives 10N+d.
- Appending k zeros multiplies N by 10k.
- n-digit repunit = (10n−1)/9.
- Repeated digit d for n places = d(10n−1)/9.
- Repeating a two-digit block gives 101 times that block.
- Repeating a three-digit block gives 1001 times that block.
- 1001 = 7 × 11 × 13.
- Greatest multiple ≤ N = N − remainder.
- If N is not divisible by d, next multiple = N + d − remainder.
- Number of multiples of d in [A,B] = ⌊B/d⌋ − ⌊(A−1)/d⌋.
- Product of two consecutive integers is divisible by 2.
- Product of three consecutive integers is divisible by 6.
- Product of k consecutive integers is divisible by k!.
- Sum of first n natural numbers = n(n+1)/2.
- Sum of first n odd numbers = n2.
- Sum of first n even numbers = n(n+1).
- Sum of first n squares = n(n+1)(2n+1)/6.
- Sum of first n cubes = [n(n+1)/2]2.
- Difference between consecutive squares = 2n+1.
- 0 is even and neither positive nor negative.
- 1 is neither prime nor composite.
- 2 is the only even prime.
- 4 is the smallest composite number.
Previous Year Questions (PYQs)
Practice these genuine previous-year questions on greatest and smallest numbers, distinct digits, repeated blocks, digit reversal, remainder conditions and important number properties.
KVS PRT PYQ22 February 2023 · Shift II
Q1. What is the difference between the greatest and the smallest five-digit numbers having all distinct digits?
A. 89,999
B. 88,531
C. 88,888
D. 89,998
Correct Answer: B. 88,531
Explanation:
The greatest five-digit number with distinct digits is:
98,765.
The smallest five-digit number with distinct digits is:
10,234.
Therefore:
98,765 − 10,234 = 88,531.
RRB Group D PYQ9 September 2022 · Shift II
Q2. Find the greatest four-digit number which leaves remainder 4 when divided by each of 7, 11 and 13.
A. 9999
B. 9013
C. 9009
D. 1005
Correct Answer: B. 9013
Explanation:
LCM(7, 11, 13) = 7 × 11 × 13 = 1001.
The required number has the form:
1001k + 4.
The greatest four-digit multiple of 1001 is:
1001 × 9 = 9009.
Therefore:
9009 + 4 = 9013.
UPSC CSE PYQ2023 · Preliminary Examination · CSAT
Q3. For any digits X, Y and Z, a six-digit number of the form XYZXYZ is always divisible by:
A. 7 and 11 only
B. 11 and 13 only
C. 7 and 13 only
D. 7, 11 and 13
Correct Answer: D. 7, 11 and 13
Explanation:
Let the three-digit number XYZ be N.
Then:
XYZXYZ = 1000N + N
= 1001N.
But:
1001 = 7 × 11 × 13.
Therefore every number of the form XYZXYZ is divisible by 7, 11 and 13.
OSSC CGL PYQ20 October 2024 · Preliminary Examination
Q4. The difference between a two-digit number and the number obtained by reversing its digits is 27. What is the difference between the two digits?
A. 2
B. 3
C. 4
D. Cannot be determined
Correct Answer: B. 3
Explanation:
Let the two-digit number be 10a + b.
Its reverse is 10b + a.
The difference is:
(10a + b) − (10b + a)
= 9(a − b).
Given:
9|a − b| = 27.
Therefore:
|a − b| = 3.
RRB ALP PYQ26 November 2024 · CBT-I · Shift I
Q5. The sum of a two-digit number and the number obtained by reversing its digits is 99. If the two digits differ by 5, which of the following can be the number?
A. 27
B. 83
C. 16
D. 18
Correct Answer: A. 27
Explanation:
Let the digits be a and b.
Number + reverse = 11(a + b).
Therefore:
11(a + b) = 99
a + b = 9.
Also:
|a − b| = 5.
The digits are 7 and 2.
Possible numbers are 27 and 72. Among the options, 27 is present.
SSC MTS PYQ16 September 2017 · Shift III
Q6. What is the sum of the first 15 odd natural numbers?
A. 255
B. 225
C. 235
D. 215
Correct Answer: B. 225
Explanation:
The sum of the first n odd natural numbers is:
n2.
For n = 15:
152 = 225.
Practice MCQs
Practice these exam-oriented questions on distinct digits, greatest and smallest multiples, specified remainders, repeated blocks, digit reversal, consecutive integers and standard number properties.
Practice MCQ
Q1. What is the smallest six-digit positive integer having all distinct digits?
A. 102345
B. 102354
C. 123450
D. 100234
Correct Answer: A. 102345
Explanation:
The first digit cannot be zero, so choose the smallest possible non-zero digit, 1.
Place 0 immediately after it, followed by the remaining smallest unused digits in ascending order:
1, 0, 2, 3, 4, 5 → 102345.
Practice MCQ
Q2. Using the digits 0, 2, 4, 7 and 9 exactly once, what is the greatest five-digit number that can be formed?
A. 97420
B. 97240
C. 94720
D. 97402
Correct Answer: A. 97420
Explanation:
For the greatest possible number, arrange all digits in descending order:
9, 7, 4, 2, 0.
Therefore the number is 97,420.
Practice MCQ
Q3. What is the difference between the greatest and the smallest four-digit numbers having distinct digits?
A. 8753
B. 8853
C. 8953
D. 8873
Correct Answer: B. 8853
Explanation:
Greatest four-digit number with distinct digits = 9876.
Smallest four-digit number with distinct digits = 1023.
Difference:
9876 − 1023 = 8853.
Practice MCQ
Q4. What is the greatest four-digit number exactly divisible by 23?
A. 9969
B. 9974
C. 9982
D. 9992
Correct Answer: C. 9982
Explanation:
The greatest four-digit number is 9999.
9999 = 23 × 434 + 17.
Subtract the remainder:
9999 − 17 = 9982.
Hence 9982 is the greatest four-digit multiple of 23.
Practice MCQ
Q5. What is the smallest three-digit number which leaves remainder 7 when divided by 13?
A. 104
B. 107
C. 111
D. 117
Correct Answer: C. 111
Explanation:
The smallest three-digit number is 100.
100 = 13 × 7 + 9.
We need remainder 7. Moving forward by 11 changes the residue from 9 to 7 modulo 13:
100 + 11 = 111.
Check:
111 = 13 × 8 + 7.
Practice MCQ
Q6. How many multiples of 12 lie from 100 to 500, both inclusive?
A. 31
B. 32
C. 33
D. 34
Correct Answer: C. 33
Explanation:
Number of multiples of 12 up to 500:
⌊500/12⌋ = 41.
Number of multiples up to 99:
⌊99/12⌋ = 8.
Required count:
41 − 8 = 33.
Practice MCQ
Q7. The number 456456 is necessarily divisible by which of the following?
A. 7 only
B. 11 only
C. 7 and 13 only
D. 7, 11 and 13
Correct Answer: D. 7, 11 and 13
Explanation:
456456 is obtained by repeating the three-digit block 456:
456456 = 456 × 1001.
Since:
1001 = 7 × 11 × 13,
456456 is divisible by 7, 11 and 13.
Practice MCQ
Q8. The sum of a two-digit number and its reverse is 132. If the difference between its digits is 4, what is the greater of the two possible numbers?
A. 75
B. 84
C. 93
D. 96
Correct Answer: B. 84
Explanation:
If the digits are a and b:
Number + reverse = 11(a+b).
Therefore:
11(a+b) = 132
a+b = 12.
Also:
|a−b| = 4.
The digits are 8 and 4.
The possible numbers are 84 and 48, so the greater number is 84.
Practice MCQ
Q9. What is the greatest natural number that is guaranteed to divide the product of any five consecutive positive integers?
A. 60
B. 90
C. 120
D. 240
Correct Answer: C. 120
Explanation:
The product of any k consecutive integers is divisible by k!.
For k = 5:
5! = 5 × 4 × 3 × 2 × 1 = 120.
This value is also the greatest guaranteed divisor because the product 1 × 2 × 3 × 4 × 5 itself equals 120.
Practice MCQ
Q10. What is the sum of the first 25 odd natural numbers?
A. 525
B. 575
C. 625
D. 675
Correct Answer: C. 625
Explanation:
The sum of the first n odd natural numbers is:
n2.
For n = 25:
252 = 625.
महत्तम/लघुत्तम संख्या संबंधी प्रश्न एवं महत्वपूर्ण संख्या गुण
Greatest या smallest number से जुड़े questions प्रायः place value, digits की arrangement, divisibility तथा remainder conditions पर आधारित होते हैं। इस topic में Number System के कई ऐसे महत्वपूर्ण properties भी शामिल हैं जो JSSC, SSC, Railway तथा अन्य competitive examinations में बार-बार उपयोगी होते हैं।
1. सबसे छोटा Positive Integer
सबसे छोटा positive integer है:
1
लेकिन कोई greatest positive integer नहीं होता, क्योंकि integers अनंत तक बढ़ते रहते हैं।
2. Greatest और Smallest One-Digit Positive Numbers
सबसे छोटी positive one-digit number:
1
सबसे बड़ी one-digit number:
9
3. Smallest n-Digit Positive Number
Exactly n digits वाली सबसे छोटी positive integer होती है:
10n−1
उदाहरण:
- Smallest 2-digit number = 10
- Smallest 3-digit number = 100
- Smallest 5-digit number = 10,000
4. Greatest n-Digit Number
Exactly n digits वाली greatest positive integer:
10n − 1
उदाहरण:
- Greatest 2-digit number = 99
- Greatest 3-digit number = 999
- Greatest 5-digit number = 99,999
5. Greatest और Smallest n-Digit Numbers का Difference
Difference:
(10n − 1) − 10n−1
= 9 × 10n−1 − 1
उदाहरण: Three-digit numbers के लिए:
999 − 100 = 899
6. दिए गए Digits से Greatest Number बनाना
दिए गए digits से greatest possible number बनाने के लिए सामान्यतः digits को descending order में arrange करें।
उदाहरण: Digits 3, 8, 1 और 6 से:
Greatest number = 8631
7. Non-Zero Digits से Smallest Number
यदि दिए गए digits में zero नहीं है, तो smallest number बनाने के लिए digits को ascending order में arrange करें।
उदाहरण: 7, 2, 9 और 4 से:
Smallest number = 2479
8. जब Zero उपस्थित हो
किसी multi-digit number का पहला digit zero नहीं हो सकता।
इसलिए smallest number बनाने के लिए:
- सबसे छोटा non-zero digit सबसे पहले रखें।
- उसके तुरंत बाद सभी zeros रखें।
- बाकी digits को ascending order में रखें।
9. Zero वाला उदाहरण
Digits 0, 5, 2, 8 और 0 से smallest number बनाइए।
Smallest non-zero digit = 2।
पहले 2, फिर दोनों zeros, फिर 5 और 8 रखें:
20,058
Common Mistake: 00258 को five-digit number नहीं माना जाएगा। Leading zeros digit count नहीं बढ़ाते।
10. Distinct Digits वाला Greatest Number
Distinct digits से greatest number बनाने के लिए सबसे बड़े available digits चुनें और descending order में रखें।
उदाहरण: Repetition के बिना greatest five-digit number:
98,765
11. Distinct Digits वाला Smallest Number
Distinct digits वाला smallest multi-digit number बनाते समय zero को सबसे पहले नहीं रखा जा सकता।
उदाहरण: Distinct digits वाला smallest five-digit number:
10,234
12. 0 से 9 तक सभी Digits का Exactly Once उपयोग
यदि 0 से 9 तक सभी digits का exactly once उपयोग करना हो:
Greatest = 9,876,543,210
Smallest = 1,023,456,789
13. जब Digit Repetition Allowed हो
यदि repetition allowed हो और कोई भी decimal digit उपयोग किया जा सकता हो:
- Greatest n-digit number = 99...9
- Smallest n-digit positive number = 10...0
अतः:
Greatest = 10n − 1
Smallest = 10n−1
14. Allowed Digits से Greatest Number
यदि repetition allowed हो, तो greatest allowed digit को repeat करें।
उदाहरण: केवल digits 2, 5 और 7 का उपयोग करके, repetition allowed हो, greatest four-digit number:
7777
15. Zero अनुपस्थित हो तो Smallest Allowed Number
यदि repetition allowed है और zero उपलब्ध नहीं है, तो smallest allowed digit को repeat करें।
उदाहरण: Digits 3, 6 और 8 से:
Smallest four-digit number = 3333
16. Zero उपलब्ध हो तो Smallest Allowed Number
यदि zero तथा कम-से-कम एक non-zero digit उपलब्ध हो, तो smallest non-zero digit पहले और उसके बाद zeros रखें।
उदाहरण: Digits 0, 4 और 9, repetition allowed:
Smallest five-digit number = 40,000
17. Two-Digit Number का Algebraic Representation
यदि tens digit = a और units digit = b हो, तो number:
10a + b
जहाँ:
1 ≤ a ≤ 9, 0 ≤ b ≤ 9
18. Two-Digit Number को Reverse करना
Original number:
10a + b
Reverse number:
10b + a
19. Two-Digit Number और उसके Reverse का Difference
Difference:
(10a + b) − (10b + a)
= 9(a − b)
महत्वपूर्ण गुण: किसी two-digit number और उसके reverse का difference हमेशा 9 से divisible होता है।
20. Two-Digit Number और उसके Reverse का Sum
Sum:
(10a + b) + (10b + a)
= 11(a + b)
महत्वपूर्ण गुण: किसी two-digit number और उसके reverse का sum हमेशा 11 से divisible होता है।
21. उदाहरण: Two-Digit Reversal
73 और 37 लें।
Difference:
73 − 37 = 36 = 9 × 4
Sum:
73 + 37 = 110 = 11 × 10
22. Three-Digit Number का Algebraic Representation
यदि digits a, b और c हों, तो number:
100a + 10b + c
Reverse number:
100c + 10b + a
23. Three-Digit Number और उसके Reverse का Difference
Subtract करें:
(100a + 10b + c) − (100c + 10b + a)
= 99(a − c)
महत्वपूर्ण गुण: Three-digit number और उसके reverse का difference हमेशा 99 से divisible होता है।
24. किसी Number के Right Side में Digit जोड़ना
यदि integer N के right side में digit d append किया जाए, तो नया number:
10N + d
उदाहरण: 245 के बाद 7 जोड़ें:
10 × 245 + 7 = 2457
25. Zero Append करना
एक zero append करने पर number 10 से multiply हो जाता है:
N → 10N
k zeros append करने पर:
N × 10k
26. किसी Number के पहले Digit लगाना
यदि N एक k-digit number है और उसके पहले digit d लगाया जाए, तो नया number:
d × 10k + N
उदाहरण: 253 के पहले 7 लगाएँ:
7 × 1000 + 253 = 7253
27. केवल 1 से बनी संख्या
n digits वाली संख्या:
111...111
को repunit कहा जाता है और इसका value होता है:
(10n − 1)/9
उदाहरण:
1111 = (104 − 1)/9
28. एक ही Digit को बार-बार लिखकर बनी संख्या
यदि digit d को n बार repeat किया जाए, तो number होगा:
d(10n − 1)/9
उदाहरण:
7777 = 7(104 − 1)/9
29. Two-Digit Block को दो बार Repeat करना
यदि किसी two-digit number N को लगातार दो बार लिखा जाए, तो नया four-digit number:
100N + N
= 101N
उदाहरण:
3535 = 35 × 101
30. Three-Digit Block को दो बार Repeat करना
यदि three-digit number N को दो बार लगातार लिखा जाए:
1000N + N = 1001N
और:
1001 = 7 × 11 × 13
महत्वपूर्ण गुण: हर six-digit number जिसका रूप abcabc हो, वह 7, 11 और 13 से divisible होता है।
31. उदाहरण: Repeated Three-Digit Block
Number लें:
357357
इसे लिख सकते हैं:
357357 = 357 × 1001
इसलिए यह 7, 11 और 13 से divisible है।
32. किसी Number का Greatest n-Digit Multiple
Greatest n-digit number:
G = 10n − 1
यदि G को d से divide करने पर remainder r मिलता है, तो:
Greatest n-digit multiple of d = G − r
33. उदाहरण: 37 का Greatest Four-Digit Multiple
Greatest four-digit number:
9999
अब:
9999 = 37 × 270 + 9
अतः:
Greatest four-digit multiple = 9999 − 9 = 9990
34. किसी Number का Smallest n-Digit Multiple
Smallest n-digit number:
S = 10n−1
यदि S पहले से d से divisible है, तो S ही answer है।
यदि S को d से divide करने पर non-zero remainder r मिले, तो:
Smallest n-digit multiple of d = S + (d − r)
35. उदाहरण: 37 का Smallest Four-Digit Multiple
Smallest four-digit number:
1000
अब:
1000 = 37 × 27 + 1
अगले multiple तक पहुँचने के लिए:
37 − 1 = 36
जोड़ें।
इसलिए:
1000 + 36 = 1036
36. N से अधिक न होने वाला Greatest Multiple
यदि N को d से divide करने पर remainder r मिले, तो:
Greatest multiple of d ≤ N = N − r
उदाहरण: 347 को 12 से divide करने पर remainder 11 मिलता है।
अतः:
Greatest multiple of 12 not exceeding 347 = 347 − 11 = 336
37. N से कम न होने वाला Smallest Multiple
यदि N को d से divide करने पर remainder r मिले:
- यदि r = 0, तो answer = N।
- यदि r ≠ 0, तो d − r जोड़ें।
अतः:
Next multiple = N + (d − r)
38. उदाहरण: 500 से बड़ा या बराबर 17 का Smallest Multiple
500 को 17 से divide करें:
500 = 17 × 29 + 7
Required addition:
17 − 7 = 10
अतः:
Required number = 500 + 10 = 510
39. Given Remainder वाला Greatest n-Digit Number
यदि greatest n-digit number चाहिए जो divisor d से divide होने पर remainder r छोड़े, तो पहले:
G = 10n − 1
लें। फिर G को नीचे adjust करें ताकि final remainder r हो जाए।
Equivalent method: G − r निकालें, उसे d से divide करने पर जो remainder मिले, उतना G से subtract करें।
40. उदाहरण: 17 से Divide करने पर Remainder 5 छोड़ने वाला Greatest Three-Digit Number
Greatest three-digit number:
999
हमें चाहिए:
N ≡ 5 (mod 17)
अब:
999 − 5 = 994
994 को 17 से divide करने पर remainder 8 मिलता है।
अतः:
N = 999 − 8 = 991
Check:
991 = 17 × 58 + 5
41. Given Remainder वाला Smallest n-Digit Number
Smallest n-digit boundary:
S = 10n−1
से शुरू करें और उसे इतना बढ़ाएँ कि required remainder प्राप्त हो।
42. उदाहरण: 13 से Divide करने पर Remainder 7 छोड़ने वाला Smallest Three-Digit Number
Smallest three-digit number:
100
अब:
100 = 13 × 7 + 9
Required remainder = 7 है।
Current remainder = 9 है।
Required increase:
7 − 9 ≡ 11 (mod 13)
इसलिए:
100 + 11 = 111
Check:
111 = 13 × 8 + 7
43. किसी Range में Multiples की संख्या
d के positive multiples जो N से अधिक नहीं हैं, उनकी संख्या:
⌊N/d⌋
इसलिए A से B inclusive तक d के multiples की संख्या:
⌊B/d⌋ − ⌊(A−1)/d⌋
44. उदाहरण: 100 से 500 तक 7 के Multiples
500 तक 7 के multiples:
⌊500/7⌋ = 71
99 तक 7 के multiples:
⌊99/7⌋ = 14
अतः required count:
71 − 14 = 57
45. Consecutive Integers
Consecutive integers को इस प्रकार represent किया जा सकता है:
n, n+1, n+2, ...
दो consecutive integers:
n और n+1
46. दो Consecutive Integers का Product
दो consecutive integers में से एक अवश्य even होगा।
इसलिए:
n(n+1) हमेशा 2 से divisible होता है।
47. तीन Consecutive Integers का Product
तीन consecutive integers में:
- कम-से-कम एक number 2 से divisible होगा।
- ठीक एक number 3 से divisible होगा।
इसलिए:
n(n+1)(n+2) हमेशा 6 से divisible होता है।
48. चार Consecutive Integers का Product
किसी भी four consecutive integers का product हमेशा:
4! = 24
से divisible होता है।
49. General Consecutive-Integer Property
किसी भी k consecutive integers का product हमेशा:
k!
से divisible होता है।
महत्वपूर्ण गुण: Product of k consecutive integers is always divisible by k!.
50. First n Natural Numbers का Sum
1 + 2 + 3 + ... + n = n(n+1)/2
उदाहरण:
1 + 2 + ... + 100 = 100 × 101 / 2 = 5050
51. First n Odd Numbers का Sum
1 + 3 + 5 + ... + (2n−1) = n2
महत्वपूर्ण गुण: First n odd numbers का sum हमेशा perfect square होता है।
52. First n Even Numbers का Sum
2 + 4 + 6 + ... + 2n = n(n+1)
53. First n Natural Numbers के Squares का Sum
12 + 22 + ... + n2 = n(n+1)(2n+1)/6
54. First n Natural Numbers के Cubes का Sum
13 + 23 + ... + n3 = [n(n+1)/2]2
महत्वपूर्ण गुण: First n natural numbers के cubes का sum, first n natural numbers के sum के square के बराबर होता है।
55. Consecutive Squares का Difference
(n+1)2 − n2
= n2 + 2n + 1 − n2
= 2n + 1
इसलिए consecutive perfect squares का difference हमेशा odd होता है।
56. हर Integer या तो Even है या Odd
हर integer को exactly एक रूप में लिखा जा सकता है:
2k
या:
2k + 1
ये क्रमशः even और odd integers को represent करते हैं।
57. Modulo 3 के तीन Possible Forms
हर integer exactly इनमें से किसी एक form में होता है:
3k, 3k+1, 3k+2
यह representation proofs और remainder-based questions में उपयोगी है।
58. महत्वपूर्ण Special Number Facts
- 0 एक even number है।
- 0 न positive है और न negative।
- 1 न prime है और न composite।
- 2 smallest prime number है।
- 2 ही only even prime number है।
- 4 smallest composite number है।
59. Sum और Product की Parity Rules
| Operation | Result |
|---|
| Even + Even | Even |
| Odd + Odd | Even |
| Even + Odd | Odd |
| Even × Any Integer | Even |
| Odd × Odd | Odd |
60. Difference की Parity Rules
- Even − Even = Even
- Odd − Odd = Even
- Even − Odd = Odd
- Odd − Even = Odd
61. Square की Parity
यदि n even है, तो n2 भी even होगा।
यदि n odd है, तो n2 भी odd होगा।
अतः:
n और n2 की parity हमेशा समान होती है।
62. Odd Square को 8 से Divide करने पर Remainder
हर odd integer को 2k+1 के रूप में लिखा जा सकता है।
एक महत्वपूर्ण standard property है:
Odd square ≡ 1 (mod 8)
उदाहरण:
- 32 = 9 → 8 से divide करने पर remainder 1
- 52 = 25 → remainder 1
- 72 = 49 → remainder 1
63. Greatest/Smallest Questions की Exam Strategy
- सबसे पहले upper या lower boundary पहचानें।
- Check करें कि digit repetition allowed है या नहीं।
- Smallest number बनाते समय zero को सावधानी से handle करें।
- Divisibility condition हो तो boundary के remainder का उपयोग करें।
- Greatest valid number के लिए सामान्यतः upper boundary से नीचे जाएँ।
- Smallest valid number के लिए lower boundary से ऊपर जाएँ।
- Specified remainder वाले questions में numbers को एक-एक करके test करने के बजाय residue relation का उपयोग करें।
64. परीक्षा में होने वाली सामान्य गलतियाँ
गलती 1: Smallest n-digit positive number = 10n−1 होता है। Leading zero का उपयोग करके smaller n-digit number नहीं बनाया जा सकता।
गलती 2: Greatest n-digit number = 10n−1 होता है।
गलती 3: Zero उपस्थित हो तो smallest multi-digit number के सबसे पहले zero न रखें।
गलती 4: Distinct digits वाले question में repetition allowed न हो तो किसी digit का दोबारा उपयोग न करें।
गलती 5: किसी number से छोटा greatest multiple पाने के लिए उसका remainder subtract करें।
गलती 6: किसी non-multiple से बड़ा next multiple पाने के लिए divisor − remainder जोड़ें।
गलती 7: Two-digit number का algebraic form 10a+b होता है, केवल a+b नहीं।
गलती 8: Two-digit number और उसके reverse का difference 9 से divisible होता है।
गलती 9: abcabc रूप की संख्या = 1001 × abc होती है और 7, 11 तथा 13 से divisible होती है।
गलती 10: k consecutive integers का product k! से divisible होता है; इसका अर्थ यह नहीं कि product हमेशा k! के बराबर होगा।
65. त्वरित पुनरावृत्ति
- Smallest n-digit positive number = 10n−1।
- Greatest n-digit number = 10n−1।
- Greatest number बनाने के लिए digits को descending order में arrange करें।
- Zero न हो तो smallest number के लिए ascending order उपयोग करें।
- Zero उपस्थित हो तो smallest non-zero digit पहले, फिर zeros रखें।
- Greatest five-digit number with distinct digits = 98765।
- Smallest five-digit number with distinct digits = 10234।
- Two-digit number = 10a+b।
- Reverse = 10b+a।
- Two-digit number और उसके reverse का difference 9 से divisible होता है।
- Two-digit number और उसके reverse का sum 11 से divisible होता है।
- Three-digit number = 100a+10b+c।
- Three-digit number और उसके reverse का difference = 99(a−c)।
- N के right side में digit d जोड़ने पर नया number = 10N+d।
- k zeros append करने पर number 10k से multiply होता है।
- n-digit repunit = (10n−1)/9।
- Digit d को n बार repeat करने पर number = d(10n−1)/9।
- Two-digit block को दो बार repeat करने पर number = 101 × block।
- Three-digit block को दो बार repeat करने पर number = 1001 × block।
- 1001 = 7 × 11 × 13।
- Greatest multiple ≤ N = N − remainder।
- यदि N, d का multiple नहीं है, तो next multiple = N + d − remainder।
- [A,B] में d के multiples की संख्या = ⌊B/d⌋ − ⌊(A−1)/d⌋।
- दो consecutive integers का product 2 से divisible होता है।
- तीन consecutive integers का product 6 से divisible होता है।
- k consecutive integers का product k! से divisible होता है।
- 1 + 2 + ... + n = n(n+1)/2।
- First n odd numbers का sum = n2।
- First n even numbers का sum = n(n+1)।
- First n squares का sum = n(n+1)(2n+1)/6।
- First n cubes का sum = [n(n+1)/2]2।
- Consecutive squares का difference = 2n+1।
- 0 even है और न positive है न negative।
- 1 neither prime nor composite है।
- 2 only even prime है।
- 4 smallest composite number है।
पिछले वर्षों में पूछे गए प्रश्न (PYQs)
Greatest/Smallest Numbers, distinct digits, repeated blocks, digit reversal, remainder conditions तथा महत्वपूर्ण number properties पर आधारित इन previous-year questions का अभ्यास करें। उत्तर एवं व्याख्या देखने से पहले प्रत्येक प्रश्न स्वयं हल करने का प्रयास करें।
KVS PRT PYQ22 फरवरी 2023 · Shift II
प्रश्न 1. सभी digits अलग-अलग होने वाली greatest तथा smallest five-digit numbers का difference कितना है?
A. 89,999
B. 88,531
C. 88,888
D. 89,998
सही उत्तर: B. 88,531
व्याख्या:
Distinct digits वाली greatest five-digit number:
98,765
Distinct digits वाली smallest five-digit number:
10,234
अतः difference:
98,765 − 10,234 = 88,531।
RRB Group D PYQ9 सितंबर 2022 · Shift II
प्रश्न 2. Greatest four-digit number ज्ञात करें जो 7, 11 तथा 13 में से प्रत्येक से divide करने पर remainder 4 छोड़े।
A. 9999
B. 9013
C. 9009
D. 1005
सही उत्तर: B. 9013
व्याख्या:
LCM(7, 11, 13) = 7 × 11 × 13 = 1001।
Required number का form होगा:
1001k + 4।
1001 का greatest four-digit multiple:
1001 × 9 = 9009।
इसलिए required number:
9009 + 4 = 9013।
UPSC CSE PYQ2023 · Preliminary Examination · CSAT
प्रश्न 3. किसी भी digits X, Y और Z के लिए XYZXYZ के रूप की six-digit number हमेशा निम्न में से किससे divisible होगी?
A. केवल 7 और 11
B. केवल 11 और 13
C. केवल 7 और 13
D. 7, 11 और 13
सही उत्तर: D. 7, 11 और 13
व्याख्या:
मान लें three-digit number XYZ = N।
तब:
XYZXYZ = 1000N + N
= 1001N।
लेकिन:
1001 = 7 × 11 × 13।
इसलिए XYZXYZ के रूप की प्रत्येक संख्या 7, 11 और 13 से divisible होगी।
OSSC CGL PYQ20 अक्टूबर 2024 · Preliminary Examination
प्रश्न 4. किसी two-digit number और उसके digits reverse करने पर प्राप्त number का difference 27 है। दोनों digits का difference कितना है?
A. 2
B. 3
C. 4
D. निर्धारित नहीं किया जा सकता
सही उत्तर: B. 3
व्याख्या:
मान लें number = 10a + b।
Reverse = 10b + a।
Difference:
(10a + b) − (10b + a)
= 9(a − b)।
Given difference = 27।
अतः:
9|a − b| = 27
|a − b| = 3।
RRB ALP PYQ26 नवंबर 2024 · CBT-I · Shift I
प्रश्न 5. किसी two-digit number तथा उसके reversed number का sum 99 है। यदि दोनों digits का difference 5 है, तो निम्न में से कौन-सी संख्या हो सकती है?
A. 27
B. 83
C. 16
D. 18
सही उत्तर: A. 27
व्याख्या:
Digits a और b मान लें।
Number + Reverse = 11(a + b)।
इसलिए:
11(a + b) = 99
a + b = 9।
साथ ही:
|a − b| = 5।
इससे digits 7 और 2 मिलते हैं।
Possible numbers = 27 तथा 72। Options में 27 उपलब्ध है।
SSC MTS PYQ16 सितंबर 2017 · Shift III
प्रश्न 6. First 15 odd natural numbers का sum कितना है?
A. 255
B. 225
C. 235
D. 215
सही उत्तर: B. 225
व्याख्या:
First n odd natural numbers का sum होता है:
n2।
n = 15 रखने पर:
152 = 225।
अभ्यास प्रश्न (Practice MCQs)
Distinct digits, greatest/smallest multiples, specified remainders, repeated blocks, digit reversal, consecutive integers तथा standard number properties पर आधारित इन exam-oriented questions का अभ्यास करें।
Practice MCQ
प्रश्न 1. सभी digits अलग-अलग होने वाली smallest six-digit positive integer कौन-सी है?
A. 102345
B. 102354
C. 123450
D. 100234
सही उत्तर: A. 102345
व्याख्या:
First digit zero नहीं हो सकता। इसलिए smallest possible non-zero digit 1 को पहले रखें।
इसके बाद 0 और फिर smallest unused digits ascending order में रखें:
1, 0, 2, 3, 4, 5 → 102345।
Practice MCQ
प्रश्न 2. Digits 0, 2, 4, 7 और 9 का exactly once उपयोग करके greatest five-digit number कौन-सी बनाई जा सकती है?
A. 97420
B. 97240
C. 94720
D. 97402
सही उत्तर: A. 97420
व्याख्या:
Greatest number बनाने के लिए digits को descending order में रखें:
9, 7, 4, 2, 0।
अतः required number = 97,420।
Practice MCQ
प्रश्न 3. Distinct digits वाली greatest तथा smallest four-digit numbers का difference कितना है?
A. 8753
B. 8853
C. 8953
D. 8873
सही उत्तर: B. 8853
व्याख्या:
Greatest four-digit number with distinct digits = 9876।
Smallest four-digit number with distinct digits = 1023।
Difference:
9876 − 1023 = 8853।
Practice MCQ
प्रश्न 4. 23 से exactly divisible greatest four-digit number कौन-सी है?
A. 9969
B. 9974
C. 9982
D. 9992
सही उत्तर: C. 9982
व्याख्या:
Greatest four-digit number = 9999।
9999 = 23 × 434 + 17।
Remainder subtract करें:
9999 − 17 = 9982।
इसलिए 9982, 23 का greatest four-digit multiple है।
Practice MCQ
प्रश्न 5. 13 से divide करने पर remainder 7 छोड़ने वाली smallest three-digit number कौन-सी है?
A. 104
B. 107
C. 111
D. 117
सही उत्तर: C. 111
व्याख्या:
Smallest three-digit number = 100।
100 = 13 × 7 + 9।
हमें remainder 7 चाहिए। Modulo 13 में 9 से 7 तक जाने के लिए 11 जोड़ना होगा।
100 + 11 = 111।
Check:
111 = 13 × 8 + 7।
Practice MCQ
प्रश्न 6. 100 से 500 तक, दोनों endpoints सहित, 12 के कितने multiples हैं?
A. 31
B. 32
C. 33
D. 34
सही उत्तर: C. 33
व्याख्या:
500 तक 12 के multiples:
⌊500/12⌋ = 41।
99 तक 12 के multiples:
⌊99/12⌋ = 8।
Required count:
41 − 8 = 33।
Practice MCQ
प्रश्न 7. संख्या 456456 निश्चित रूप से निम्न में से किससे divisible है?
A. केवल 7
B. केवल 11
C. केवल 7 और 13
D. 7, 11 और 13
सही उत्तर: D. 7, 11 और 13
व्याख्या:
456456 में three-digit block 456 दो बार repeat हुआ है।
456456 = 456 × 1001।
और:
1001 = 7 × 11 × 13।
इसलिए 456456, 7, 11 और 13 तीनों से divisible है।
Practice MCQ
प्रश्न 8. किसी two-digit number तथा उसके reverse का sum 132 है। यदि दोनों digits का difference 4 है, तो दोनों possible numbers में greater number कौन-सी है?
A. 75
B. 84
C. 93
D. 96
सही उत्तर: B. 84
व्याख्या:
Digits a और b मान लें।
Number + Reverse = 11(a+b)।
इसलिए:
11(a+b) = 132
a+b = 12।
साथ ही:
|a−b| = 4।
Digits = 8 और 4।
Possible numbers = 84 और 48। इसलिए greater number = 84।
Practice MCQ
प्रश्न 9. किसी भी पाँच consecutive positive integers के product को हमेशा divide करने वाली greatest natural number कौन-सी है?
A. 60
B. 90
C. 120
D. 240
सही उत्तर: C. 120
व्याख्या:
किसी भी k consecutive integers का product k! से divisible होता है।
k = 5 के लिए:
5! = 5 × 4 × 3 × 2 × 1 = 120।
यह greatest guaranteed divisor भी है, क्योंकि 1 × 2 × 3 × 4 × 5 का product स्वयं 120 है।
Practice MCQ
प्रश्न 10. First 25 odd natural numbers का sum कितना है?
A. 525
B. 575
C. 625
D. 675
सही उत्तर: C. 625
व्याख्या:
First n odd natural numbers का sum होता है:
n2।
n = 25 रखने पर:
252 = 625।