Problem

A student was asked to prove, for $p\in\mathbb{N}$, that

“if $p^3$ is a multiple of 3, then $p$ must be a multiple of 3”

The start of the student’s proof by contradiction is shown in the box below.

(a) Show the calculations and statements that are required to complete the proof.

(b) Hence prove, by contradiction, that $\sqrt[3]{3}$ is an irrational number.