mathschallenge.net

Problem

Given that $a$ and $b$ are positive integers, find the conditions for which the equation $a$ $b$ = $c$ has a solution.

Solution

From $a$ = $b$ + $c$, square both sides, $a$ = $b$² + 2$b$$c$ + $c$.

Rearranging we get,

$a$ $b$2 $c$ 2$b$

= $c$.

As the left hand side is rational, $c$ must be rational.

Let $c$=$x$/$y$, where HCF($x$, $y$)=1.

Squaring, $c$=$x$²/$y$², $cy$²=$x$².

As the left hand side divides by y² and HCF($x$², $y$²)=1, the right hand side will only divide by $y$² if $y$²=1. Hence $c$=$x$² must be a perfect square.

Furthermore, if $c$ is a perfect square, $a$ = $b$ + $c$ will be integer, so $a$ must also be a perfect square.