When the three-digit positive integer $N$ is divided by 10,11 or 12 , the remainder is 7 . What is the sum of the digits of $N$ ?
(A) 15
(B) 17
(C) 23
(D) 11
(E) 19

Part C: Each correct answer is worth 8.
Solution 1

When $N$ is divided by 10,11 or 12 , the remainder is 7 .

This means that $M=N-7$ is divisible by each of 10,11 and 12 .

Since $M$ is divisible by each of 10,11 and 12 , then $M$ is divisible by the least common multiple of 10,11 and 12 .

Since $10=2 \times 5,12=2 \times 2 \times 3$, and 11 is prime, then the least common multiple of 10,11 and 12 is $2 \times 2 \times 3 \times 5 \times 11=660$. (To find the least common multiple, we compute the product of the highest powers of each of the prime factors that occur in the given numbers.)

Since $M$ is divisible by 660 and $N=M+7$ is a three-digit positive integer, then $M$ must equal 660. (The next largest multiple of 660 is 1320 .)

Therefore, $N=M+7=667$, and so the sum of the digits of $N$ is $6+6+7=19$.

## Solution 2

When $N$ is divided by 10,11 or 12 , the remainder is 7 .

This means that $M=N-7$ is divisible by each of 10,11 and 12 .

Since $M$ is divisible by each of 10 and 11 , then $M$ must be divisible by 110 .

We test the first few multiples of 110 until we obtain one that is divisible by 12 .

The integers 110, 220, 330, 440, and 550 are not divisible by 12 , but 660 is.

Therefore, $M$ could be 660 . (This means that $M$ must be 660 .)

Finally, $N=M+7=667$, and so the sum of the digits of $N$ is $6+6+7=19$.

ANsWER: (E)