Multiple choice

What is the sum of digits of the least number, which when divided by 15, 18 and 24 leaves the same remainder 8 in each case and is also divisible by 13?

  1. 18

  2. 16

  3. 19

  4. 17
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D Correct answer
Explanation

The number must be of the form LCM(15, 18, 24) * k + 8. The LCM is 360, so the number is 360k + 8. We need 360k + 8 to be divisible by 13; testing values of k, 360k + 8 = 27k + 8 (mod 13), which simplifies to k + 8 = 0 (mod 13), so k = 5. The number is 360(5) + 8 = 1808, and the sum of digits is 1+8+0+8 = 17.

AI explanation

The least common multiple of 15, 18, and 24 is 360, so the required number must be of the form 360k plus 8. Testing values for k, when k is 2, the number is 728, which is perfectly divisible by 13. The sum of the digits of 728 is 7 plus 2 plus 8, resulting in 17.

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