If n is an integer such that 1nn352 is a six-digit number exactly divisible by 24, what will be the sum of the possible values of n?

To find the possible values of n, we need to understand the rules of divisibility by 24. A number is divisible by 24 if it is divisible by both 3 and 8.

1. Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. In this case, the sum of the digits is 1 + n + n + 3 + 5 + 2 = 11 + 2n. For this to be divisible by 3, 2n must be 1 or 4 or 7. So, the possible values of n are 0.5, 2, 3.5. But n is an integer, so n can only be 2.

2. Divisibility by 8: A number is divisible by 8 if the number formed by the last three digits is divisible by 8. In this case, the last three digits form the number 35n. For this to be divisible by 8, n must be 2 or 6.

Therefore, the only possible value of n that satisfies both conditions is 2. So, the sum of the possible values of n is 2.