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Slovenia — National Math Olympiad

Number theory Difficulty 4.8 AIME Prove it Slovenia

Find all non-zero integers $a$ , different from $4$ , such that $\frac{a}{a-4} + \frac{2}{a}$ is an integer as well.

Solution

If $\frac{a}{a-4} + \frac{2}{a} = \frac{a^2+2a-8}{a(a-4)}$ is an integer, then $a(a-4)$ divides $a^2+2a-8$ . So, $a$ divides $a^2+2a-8$ , and $a$ divides $8$ . All integer divisors of $8$ different from $4$ are $1$ , $2$ , $8$ , $-1$ , $-2$ , $-4$ and $-8$ . We check all seven cases and see that the value of the expression is an integer only when $a = 2$ or $a = -4$ .

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