Multiple choice

What is the sum of the digits of the least number which when divided by 15, 18 and 36 leaves the same remainder 9 in each case and is divisible by II?

  1. 16

  2. 17

  3. 15

  4. 18
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D Correct answer
Explanation

The number is 9 more than a multiple of 15, 18, and 36, so it has the form 36k + 9. Requiring divisibility by 11 gives the least suitable number as 1089, whose digit sum is 18.

AI explanation

The given number must leave a remainder of 9 when divided by 15, 18, and 36, meaning it must be of the form L + 9 where L is the least common multiple of those divisors. The LCM of 15, 18, and 36 is 180, so the number must be a multiple of 180 plus 9, making it 189 plus multiples of 180. The general form of the number is 189k, and because the number must also be divisible by 11, we test values of k. When k equals 1, the number 189 divided by 11 leaves a remainder of 2, so we solve 2k divided by 11 to find the remainder, meaning k must be a multiple of 11 to yield a zero remainder. Setting k to 11 gives the least such number as 189 multiplied by 11, which is 2079. The sum of the digits of 2079 is 2 + 0 + 7 + 9, which equals 18.

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