Multiple choice

What is the sum of the digits of the least number, which when divided by 12, 16 and 54 leaves the same remainder 7 in each case, and is also completely divisible by 13?

  1. 36

  2. 16

  3. 9

  4. 27
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

LCM(12, 16, 54) = 432. Number = 432k + 7. We need 432k + 7 to be divisible by 13. 432 mod 13 = 3. So (3k + 7) mod 13 = 0. For k=2, 6+7=13 (divisible). Number = 432(2) + 7 = 864 + 7 = 871. Sum of digits = 8+7+1 = 16.

AI explanation

Subtract the common remainder 7 from the divisors 12, 16 and 54, and find the least common multiple (LCM) of the resulting numbers. The LCM of 5, 9 and 47 is 5 times 9 times 47, which equals 2115. We need the multiple of 2115 that leaves a remainder of 7, so we check the unit digit of numbers formed by adding 7 to multiples of 2115; testing 3 times 2115 plus 7 gives 6352, which is completely divisible by 13. The sum of the digits of 6352 is 6 plus 3 plus 5 plus 2, which equals 16. The sum of the digits of this least number is 16.

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