Number System MCQ – Part 2 - Topic Guide and MCQs

1. Which pair consists of two composite numbers that are co-prime?

   1. (21, 35)
   2. (27, 40)
   3. (22, 33)
   4. (26, 39)
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   Answer: (B) (27, 40)

   Explanation: 27 = 3³ and 40 = 2³ × 5 have no common prime factor, so their HCF is 1. Each other pair has a common factor greater than 1.

2. Find the remainder when 11^57 is divided by 15.

   1. 1
   2. 4
   3. 14
   4. 11
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   Answer: (D) 11

   Explanation: 11² = 121 leaves remainder 1 on division by 15. Since 57 is odd, 11^57 leaves the same remainder as 11, namely 11.

3. What is the least positive integer by which 864 must be divided to obtain a perfect cube?

   1. 4
   2. 2
   3. 3
   4. 6
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   Answer: (A) 4

   Explanation: 864 = 2^5 × 3³. Divide by 2² = 4 to leave 2³ × 3³ = 216, a perfect cube.

4. A number leaves remainder 8 when divided by 13. What remainder will three times the number leave when divided by 13?

   1. 8
   2. 5
   3. 11
   4. 2
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   Answer: (C) 11

   Explanation: Three times the number has the same remainder as 3 × 8 = 24. Thus the remainder is 24 − 13 = 11.

5. What are the last two digits of 7^202?

   1. 07
   2. 49
   3. 43
   4. 01
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   Answer: (B) 49

   Explanation: 7^4 = 2401 ends in 01, so the last two digits repeat every 4 powers. Since 202 leaves remainder 2 on division by 4, the last two digits are those of 7², namely 49.

6. The six-digit number 83x52y is divisible by 55. Find the greatest possible value of x − y.

   1. 8
   2. 4
   3. 9
   4. 5
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   Answer: (C) 9

   Explanation: Divisibility by 5 gives y = 0 or 5. The alternating-sum test for 11 gives x − y + 2 as a multiple of 11: the valid pairs are (9, 0) and (3, 5), so the greatest difference is 9.

7. If p is prime and p² has exactly three positive factors, how many positive factors does p^5 have?

   1. 6
   2. 5
   3. 10
   4. 7
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   Answer: (A) 6

   Explanation: For a prime power p^k, the number of positive factors is k + 1. Hence p^5 has 5 + 1 = 6 factors.

8. Find the sum of the squares of the first 12 natural numbers.

   1. 608
   2. 676
   3. 600
   4. 650
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   Answer: (D) 650

   Explanation: Use n(n + 1)(2n + 1) ÷ 6: 12 × 13 × 25 ÷ 6 = 650.

9. What is the greatest three-digit number that leaves remainder 7 when divided by each of 16 and 24?

   1. 991
   2. 967
   3. 960
   4. 983
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   Answer: (B) 967

   Explanation: LCM of 16 and 24 is 48. The greatest multiple of 48 not exceeding 999 − 7 = 992 is 960, so the number is 960 + 7 = 967.

10. What is the unit digit of 4^38 × 9^27?

    1. 6
    2. 9
    3. 4
    4. 1
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    Answer: (C) 4

    Explanation: An even power of 4 ends in 6, and an odd power of 9 ends in 9. Their product ends in the unit digit of 6 × 9 = 54, which is 4.

11. How many digits x make the five-digit number 43x76 divisible by 3?

    1. 3
    2. 2
    3. 4
    4. 5
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    Answer: (A) 3

    Explanation: The digit sum is 20 + x. It is divisible by 3 for x = 1, 4 and 7, giving 3 possible digits.

12. The 6th and 14th terms of an arithmetic progression are 29 and 69. Find its first term.

    1. 5
    2. 9
    3. −1
    4. 4
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    Answer: (D) 4

    Explanation: The common difference is (69 − 29) ÷ (14 − 6) = 5. The first term is 29 − 5 × 5 = 4.

13. How many integers between 200 and 500, including both endpoints, are divisible by 7 but not by 5?

    1. 35
    2. 34
    3. 43
    4. 9
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    Answer: (B) 34

    Explanation: There are 71 − 28 = 43 multiples of 7 and 14 − 5 = 9 multiples of 35 in this interval. Hence the required count is 43 − 9 = 34.

14. Find the value of 48 × 52 + 2².

    1. 2496
    2. 2504
    3. 2500
    4. 2304
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    Answer: (C) 2500

    Explanation: 48 × 52 = (50 − 2)(50 + 2) = 50² − 2². Adding 2² leaves 50² = 2500.

15. A three-digit number has hundreds digit 5 and units digit 3. Interchanging its hundreds and units digits reduces it by how much?

    1. 198
    2. 180
    3. 200
    4. 18
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    Answer: (A) 198

    Explanation: The tens digit cancels. The reduction is 99 × (5 − 3) = 198.

16. Find the remainder when 2^50 + 3^50 is divided by 7.

    1. 1
    2. 4
    3. 0
    4. 6
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    Answer: (D) 6

    Explanation: Since 2³ leaves remainder 1, 2^50 leaves remainder 4. Since 3^6 leaves remainder 1, 3^50 leaves remainder 2; the total remainder is 4 + 2 = 6.

17. What is the smallest integer greater than 1 that divides both 2^9 − 1 and 2^6 − 1 exactly?

    1. 3
    2. 9
    3. 7
    4. 21
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    Answer: (C) 7

    Explanation: 2^9 − 1 = 511 = 7 × 73 and 2^6 − 1 = 63 = 7 × 9. Their HCF is 7, whose smallest divisor greater than 1 is 7.

18. What are the last two digits of 25^19 + 75^20?

    1. 00
    2. 50
    3. 25
    4. 75
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    Answer: (B) 50

    Explanation: Every positive power of 25 ends in 25. An even positive power of 75 ends in 25, so the last two digits of the sum are 25 + 25 = 50.

19. How many trailing zeros are there in the product 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10?

    1. 2
    2. 1
    3. 3
    4. 5
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    Answer: (A) 2

    Explanation: Only 5 and 10 supply factors of 5, while factors of 2 are plentiful. Thus there are 2 trailing zeros.

20. A number is divided by 4, and the resulting quotient is divided by 6. The respective remainders are 3 and 5. What remainder does the original number leave when divided by 24?

    1. 8
    2. 17
    3. 19
    4. 23
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    Answer: (D) 23

    Explanation: Write the first quotient as 6k + 5. Then the original number is 4(6k + 5) + 3 = 24k + 23, so the remainder is 23.

21. How many positive factors of 2^8 × 3^6 × 5^4 are perfect squares?

    1. 30
    2. 60
    3. 120
    4. 315
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    Answer: (B) 60

    Explanation: A square factor must have even prime exponents. There are 5 choices for the exponent of 2, 4 for that of 3 and 3 for that of 5, giving 5 × 4 × 3 = 60.

22. A square courtyard has area 2,025 m². If its side is increased by 3 m, what is the new area?

    1. 2034 m²
    2. 2160 m²
    3. 2304 m²
    4. 2401 m²
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    Answer: (C) 2304 m²

    Explanation: The original side is √2025 = 45 m. The new area is (45 + 3)² = 48² = 2304 m².

23. A three-digit number has digit sum 16. Is it greater than 700? Statement I: Its hundreds digit is twice its units digit. Statement II: It exceeds the number formed by reversing its digits by 396. Choose the correct data-sufficiency conclusion.

    1. Both statements together are needed
    2. Only statement I is sufficient
    3. Only statement II is sufficient
    4. Either statement alone is sufficient
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    Answer: (A) Both statements together are needed

    Explanation: Statement I allows 673 and 844; statement II allows 682 and 763, so neither alone settles the comparison. Together, h = 2u and h − u = 396 ÷ 99 = 4 give u = 4, h = 8 and t = 16 − 8 − 4 = 4, so the number is 844.

24. What is the smallest positive integer that must be added to 4,321 to make the result divisible by both 8 and 9?

    1. 7
    2. 8
    3. 1
    4. 71
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    Answer: (D) 71

    Explanation: LCM of 8 and 9 is 72, and 4321 = 72 × 60 + 1. Add 72 − 1 = 71 to reach the next multiple.

25. Using each of the digits 1, 3, 5 and 7 exactly once, find the difference between the greatest and smallest four-digit numbers formed.

    1. 6084
    2. 6174
    3. 6264
    4. 5994
    Show answer

    Answer: (B) 6174

    Explanation: The greatest number is 7531 and the smallest is 1357. Their difference is 7531 − 1357 = 6174.

Number System MCQ – Part 2 is factual rather than analytical: you either recall it in seconds or you lose the mark to somebody who did. That makes it one of the cheapest sections to convert - a handful of short sittings usually settles the whole cluster.

Expect one-line recall questions - definitions, stand-for expansions, identifiers and the ordering of things. Practise the wording as well as the concept: these papers test terminology, and the set above uses the same four-option shape they do.

Past papers reuse clusters like this almost verbatim across years and posts. Run the same set until every distractor is familiar - previous-year style questions stop being recall and start being free marks.