CONO-SUR 2014 Problem 2

A pair of positive integers $(a,b)$ is called charrua if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$ , then the pair is called non-charrua.

a) Prove that there are infinite non-charrua pairs. b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is charrua.