# 1959 AHSME Problems/Problem 14

Given the set $S$ whose elements are zero and the even integers, positive and negative. Of the five operations applied to any pair of elements: (1) addition (2) subtraction (3) multiplication (4) division (5) finding the arithmetic mean (average), those elements that only yield elements of $S$ are: $\textbf{(A)}\ \text{all} \qquad\textbf{(B)}\ 1,2,3,4\qquad\textbf{(C)}\ 1,2,3,5\qquad\textbf{(D)}\ 1,2,3\qquad\textbf{(E)}\ 1,3,5$

## Solution

The first three listed operations, applied to even integers, all yield even integers trivially. As for the fourth operation (division) and fifth operation (average of pair), we can find pairs of even numbers that are mapped to odd integers, such as $\frac{2}{2} = 1$, and $\frac{0+2}{2} = 1$. Therefore, the operations that map even integers to even integers only are operations $1, 2, 3$, so our answer is $\boxed{\textbf{(D)}}$ and we are done.

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