Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Find the answer

For how many pairs of nonzero integers $(c, d)$ with $-2015 \leq c, d \leq 2015$ do the equations $c x=d$ and $d x=c$ both have an integer solution?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We need both $c / d$ and $d / c$ to be integers, which is equivalent to $|c|=|d|$ , or $d= \pm c$ . So there are 4030 ways to pick $c$ and 2 ways to pick $d$ , for a total of 8060 pairs.

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