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 = b² + 2bc + c.

Rearranging we get,

a b2 c 2b

= c.

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

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

Squaring, c=x²/y², cy²=x².

As the left hand side divides by y² and HCF(x², y²)=1, the right hand side will only divide by y² if y²=1. Hence c=x² 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.