real analysis - Why do we consider that $p$ & $q$ are co-primes when proving square root of a prime number is irrational?

When we prove that square root of any prime number is irrational, we assume that there exists some rational number $r=\frac{p}{q};\space p,q\in\mathbb{Z}, q\neq0$ s.t. $\frac{p^2}{q^2}=p_1$ where $p_1$ is prime and then prove by contradiction.

Why do we consider that $p$ & $q$ are co-primes?

Answer

This image is from pg-2 of Abbott's Understanding Analysis that gives the simple yet elegant answer to the question: