Number System MCQ – Part 1 - Topic Guide and MCQs

1. Which of the following is an irrational number?

   1. √18
   2. √81
   3. 0.625
   4. −13/4
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   Answer: (A) √18

   Explanation: √18 = 3√2, which is irrational; the other three values are rational.

2. If the four-digit number 52x4 is divisible by 12, find the digit x.

   1. 1
   2. 7
   3. 4
   4. 8
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   Answer: (C) 4

   Explanation: Divisibility by 4 requires x to be even. The digit sum 11 + x must also be divisible by 3, so x = 4.

3. How many positive factors of 7,560 are divisible by 12?

   1. 32
   2. 24
   3. 48
   4. 16
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   Answer: (B) 24

   Explanation: 7,560 = 2³ × 3³ × 5 × 7. For divisibility by 12, the factor must contain at least 2² × 3; the count is (3 − 2 + 1) × (3 − 1 + 1) × 2 × 2 = 24.

4. Using each of the digits 0, 2, 6 and 9 exactly once, what is the smallest four-digit even number?

   1. 2069
   2. 2690
   3. 2906
   4. 2096
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   Answer: (D) 2096

   Explanation: Start with 2 and then 0 to minimise the number. Of the remaining digits, 6 must be last to make it even, giving 2096.

5. What is the unit digit of 3^47?

   1. 7
   2. 3
   3. 9
   4. 1
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   Answer: (A) 7

   Explanation: The unit digits repeat as 3, 9, 7, 1. Since 47 leaves remainder 3 on division by 4, the unit digit is 7.

6. What is the least positive integer by which 540 must be multiplied to obtain a perfect square?

   1. 5
   2. 15
   3. 10
   4. 30
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   Answer: (B) 15

   Explanation: 540 = 2² × 3³ × 5. Multiply by 3 × 5 = 15 to make every prime exponent even.

7. Find the remainder when 7^103 is divided by 13.

   1. 7
   2. 5
   3. 12
   4. 6
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   Answer: (D) 6

   Explanation: 7³ leaves remainder 5 and 7^6 leaves remainder −1 modulo 13. Since 103 = 6 × 17 + 1, the remainder is −7 + 13 = 6.

8. A number divided by 23 gives quotient 17 and remainder 9. Find the number.

   1. 391
   2. 382
   3. 400
   4. 409
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   Answer: (C) 400

   Explanation: Dividend = 23 × 17 + 9 = 400.

9. Find the sum of all odd integers from 31 to 79, including both endpoints.

   1. 1375
   2. 1320
   3. 1430
   4. 2750
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   Answer: (A) 1375

   Explanation: There are (79 − 31) ÷ 2 + 1 = 25 terms. Sum = 25 × (31 + 79) ÷ 2 = 1375.

10. What is the smallest four-digit number that leaves remainder 5 when divided by each of 12, 18 and 30?

    1. 1005
    2. 1080
    3. 1265
    4. 1085
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    Answer: (D) 1085

    Explanation: LCM of 12, 18 and 30 is 180. The least multiple of 180 at least 995 is 1080, so the required number is 1080 + 5 = 1085.

11. What is the greatest positive integer that divides n² − 1 for every odd integer n?

    1. 4
    2. 8
    3. 16
    4. 24
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    Answer: (B) 8

    Explanation: n² − 1 = (n − 1)(n + 1), a product of consecutive even integers, so it is divisible by 8. Taking n = 3 gives 8, so no larger integer always divides it.

12. Find the sum of all positive factors of 150.

    1. 360
    2. 186
    3. 372
    4. 300
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    Answer: (C) 372

    Explanation: 150 = 2 × 3 × 5². Sum of factors = (1 + 2)(1 + 3)(1 + 5 + 25) = 3 × 4 × 31 = 372.

13. A two-digit number exceeds its reverse by 36. The sum of the squares of its digits is 80. Find the original number.

    1. 84
    2. 48
    3. 62
    4. 73
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    Answer: (A) 84

    Explanation: For digits x and y, x − y = 36 ÷ 9 = 4. Then (x + y)² = 2(x² + y²) − (x − y)² = 160 − 16 = 144, so x + y = 12 and the digits are 8 and 4.

14. What is the unit digit of 2^35 + 3^26 + 7^19?

    1. 2
    2. 0
    3. 8
    4. 6
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    Answer: (B) 0

    Explanation: The three unit digits are 8, 9 and 3 respectively. Their sum is 20, so the required unit digit is 0.

15. An arithmetic progression begins with 11, 18, 25 and 32. Find its 20th term.

    1. 151
    2. 140
    3. 133
    4. 144
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    Answer: (D) 144

    Explanation: The common difference is 7. The 20th term is 11 + (20 − 1) × 7 = 144.

16. If 7x32y is a five-digit number divisible by 72, what is the greatest possible value of x + y?

    1. 6
    2. 14
    3. 15
    4. 18
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    Answer: (C) 15

    Explanation: Divisibility by 8 gives y = 0 or 8. The digit sum 12 + x + y must be divisible by 9: the pairs (x, y) are (6, 0) and (7, 8), so the greatest sum is 15.

17. What is the least non-negative integer that must be subtracted from 950 to obtain a perfect square?

    1. 11
    2. 50
    3. 49
    4. 89
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    Answer: (B) 50

    Explanation: 30² = 900 and 31² = 961. The nearest perfect square below 950 is 900, so subtract 950 − 900 = 50.

18. How many trailing zeros are there in 125!?

    1. 31
    2. 25
    3. 30
    4. 32
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    Answer: (A) 31

    Explanation: Count factors of 5: 125 ÷ 5 = 25, 125 ÷ 25 = 5 and 125 ÷ 125 = 1. Total trailing zeros = 25 + 5 + 1 = 31.

19. Evaluate 1003² − 997².

    1. 36
    2. 6000
    3. 12000
    4. 24000
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    Answer: (C) 12000

    Explanation: Use a² − b² = (a − b)(a + b): 6 × 2000 = 12000.

20. A number divided by a positive integer gives quotient 18 and remainder 7. When the divisor is increased by 5, the quotient becomes 15 and the remainder stays 7. Find the number.

    1. 450
    2. 547
    3. 382
    4. 457
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    Answer: (D) 457

    Explanation: Let the original divisor be d. Then 18d + 7 = 15(d + 5) + 7, so 3d = 75 and d = 25; the number is 18 × 25 + 7 = 457.

21. How many positive factors does 81 have?

    1. 5
    2. 4
    3. 6
    4. 9
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    Answer: (A) 5

    Explanation: 81 = 3^4, so the number of positive factors is 4 + 1 = 5.

22. A positive integer N leaves remainder 4 when divided by 9. What remainder does N² + 3N + 2 leave when divided by 9?

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

    Explanation: Replace N by its remainder 4: 4² + 3 × 4 + 2 = 30. Dividing 30 by 9 leaves remainder 3.

23. What is the greatest five-digit number exactly divisible by 88?

    1. 99912
    2. 99968
    3. 99944
    4. 99880
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    Answer: (B) 99968

    Explanation: 99,999 = 88 × 1136 + 31. Subtract the remainder: 99,999 − 31 = 99,968.

24. Among four-digit numbers of the form 7x2y divisible by 5, 9 and 11, find the difference between the greatest and the smallest. Here x and y are digits.

    1. 450
    2. 990
    3. 545
    4. 495
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    Answer: (D) 495

    Explanation: Divisibility by 11 forces x + y = 9, which also makes the digit sum 18. Divisibility by 5 gives y = 0 or 5, producing 7920 and 7425; their difference is 495.

25. Harjit writes three consecutive odd positive integers whose sum is 195. What is the largest of them?

    1. 67
    2. 65
    3. 66
    4. 69
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    Answer: (A) 67

    Explanation: The middle integer is the average, 195 ÷ 3 = 65. The largest is 65 + 2 = 67.

Number System MCQ – Part 1 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.