### Divisibility & Induction

Food for thought

Find the smallest positive integers $a$, $b$ and $c$ such that $a^2 = 2b^3 = 3c^5.$
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 $b$?