Number System

Ladakh Police Sub-Inspector · Mathematics · 99 practice questions with answers and explanations: 45 Easy, 49 Medium, 5 Hard.

Sample questions

Sample Easy question 1

Find the smallest 3-digit number divisible by 7.

  1. 100
  2. 104
  3. 102
  4. 105
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Answer: D. 100/7 ≈ 14.3, so the next multiple, 15×7 = 105, is the smallest 3-digit multiple.

Sample Easy question 2

Find the smallest 3-digit number divisible by 9.

  1. 103
  2. 108
  3. 106
  4. 105
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Answer: B. 100/9 ≈ 11.1, so the next multiple, 12×9 = 108, is the smallest 3-digit multiple.

Sample Easy question 3

Find the largest 3-digit number divisible by 6.

  1. 992
  2. 994
  3. 996
  4. 990
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Answer: C. 999/6 = 166.5, so 166×6 = 996 is the largest 3-digit multiple.

Sample Medium question 4

Find the remainder when 5²³ is divided by 6.

  1. 0
  2. 1
  3. 3
  4. 5
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Answer: D. 5 ≡ -1 (mod 6); (-1)²³ = -1 ≡ 5 (mod 6).

Sample Medium question 5

Find the remainder when 4¹⁵ is divided by 7.

  1. 1
  2. 6
  3. 4
  4. 2
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Answer: A. Powers of 4 mod 7 cycle as 4,2,1 (period 3); 15 mod 3 = 0, giving remainder 1.

Sample Medium question 6

Find the number of distinct prime factors of 210 (= 2×3×5×7).

  1. 3
  2. 2
  3. 5
  4. 4
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Answer: D. 210 has exactly 4 distinct prime factors: 2, 3, 5, and 7.

Sample Hard question 7

Find the remainder when 3¹⁰⁰ is divided by 7.

  1. 4
  2. 6
  3. 1
  4. 2
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Answer: A. Powers of 3 mod 7 cycle as 3,2,6,4,5,1 (period 6); 100 mod 6 = 4, giving remainder 4.

Sample Hard question 8

Find the remainder when 1³ + 2³ + ... + 10³ is divided by 9.

  1. 3
  2. 0
  3. 1
  4. 2
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Answer: C. Sum of cubes 1 to 10 is 3025; 3025 mod 9 = 1.

Sample Hard question 9

Find the number of digits in 2³⁰ (use log₁₀2 ≈ 0.30103).

  1. 11
  2. 9
  3. 8
  4. 10
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Answer: D. 30 × 0.30103 ≈ 9.03; number of digits = ⌊9.03⌋ + 1 = 10.

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