Multiple choice

What is the smallest number which gives a remainder of 5 in each case when divided by 8, 10, 12 but leaves no remainder when divided by 13.

  1. 845

  2. 407

  3. 650

  4. 910

  5. 1040
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Number giving remainder 5 when divided by 8, 10, 12 is LCM(8,10,12) × k + 5 = 120k + 5. For divisibility by 13: 120k + 5 ≡ 0 (mod 13), so 120k ≡ 8 (mod 13). 120 ≡ 3 (mod 13), so 3k ≡ 8 (mod 13). Testing k = 7: 3×7 = 21 ≡ 8 (mod 13). Number = 120×7 + 5 = 845. Check: 845 ÷ 13 = 65 exactly.

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