mathschallenge.net

Problem

Given that $a$ and $b$ are positive integers, for which values is the following expression integer?

$a$/$b$ + $b$/$a$

Solution

Let $c$ = $a$/$b$ + $b$/$a$.

Suppose that HCF($a$,$b$) = $k$, such that $a$ = $kd$ and $b$ = $ke$.

Therefore, $c$ = $d$/$e$ + $e$/$d$ and HCF($d$,$e$) = 1.

Multiplying through by $d$ gives $cd$ = $d$²/$e$ + $e$. Clearly $cd$ and $e$ are integer, so $d$²/$e$ must be integer. But as HCF($d$,$e$) = 1, we deduce that $e$ = 1; and by symmetry $d$ = 1.

Hence $c$ = $d$/$e$ + $e$/$d$ = 1 + 1 = 2, and as $a$/$b$ = $d$/$e$ = 1, we further deduce that $a$ = $b$ can take on any non-zero integer value.