Find the smallest positive integers \(a\), \(b\) and \(c\) such that \[a^2 = 2b^3 = 3c^5.\]

What can you say about other solutions to this equation?

The question is really asking us to solve two simultaneous equations, so we could start by concentrating on just one. Can we find the smallest positive integers \(a\) and \(b\) such that \(a^2 = 2b^3\)? Can we find all solutions?

In the equation \(a^2 = 2b^3\), the right-hand side is even. What does that tell us about \(a\) and \(b\)?

Could \(a\) be divisible by \(3\)? Could \(b\)? Can we say anything about the prime factorisations of \(a\) and \(b\)?