The question is "Using the contrapositive prove for all integers n, if n^2 is a multiple of 5 then n is a multiple of 5". I know that the contrapositive is "if n is not a multiple of 5 then n^2 is not a multiple a 5".
I have gotten as far as writing n = 5q + r, but I am not sure where to go from here.
Any help would be appreciated.
Since $n$ is not a multiple of 5, we can assume that $r\in\{1,2,3,4\}$. Therefore $n^2=25q + 10qr + r^2$. The first two terms are multiples of 5, but not the last (since $r^2\in\{1,4,9,16\}$). Thus $n^2$ is not a multiple of 5.