In the field of cryptography, certain protocols rely on the property that specific constants are not rational. Let $S$ be a constant defined by the equation $S^2-13=0$. Given that $S>0$, prove by contradiction that $S$ is an irrational number.

(You may assume that if $n^2$ is a multiple of 13, then $n$ is also a multiple of 13 for any integer $n$.)