15) On my blackboard, there are some positive integers, not necessarily distinct. If I add them up, I get 83, if I multiply them, I get 1024. What is the smallest of the numbers written on my blackboard?
(A) 1 ,
(B) 2 ,
(C) 4 ,
(D) 8 ,
(E) 16 .
15. The answer is (A).

The starting observation is that $1024=2^{10}$ and therefore its divisors are the powers of 2, from the one with exponent 0, i.e., 1, to the one with exponent 10, i.e., 1024 itself. Thus, the numbers written on the board can only be among these. On the other hand, the sum of such numbers is 83, from which it follows that the numbers written on the board can only be: 1, 2, 4, 8, 16, 32, 64. Now observe that 83 is odd, so 1 must necessarily be one of the numbers written on the board, since any number obtained as the sum of 2, 4, 8, 16, 32, and 64 is even. Therefore, the smallest number written on the board is 1. Finally, one possibility is that the numbers written on the board are: $1,1,1,16,64$.

[Problem proposed by M. Mamino.]