Finding the factors of integer $x$ and its square

Question

What is the the theorem or property that says that $\forall{}x\in\Bbb Z$, the set of all integers, $x^2$ has the same factors as $x$, twice?

Answer 1

Well, the property may be the following: let $$ x={p_1}^{a_1}\cdot\ldots\cdot{p_n}^{a_n} $$ be the unique factorization of $x\in\Bbb Z$, $x>1$, then $$ x^2=\left({p_1}^{a_1}\cdot\ldots\cdot{p_n}^{a_n}\right)^2= {p_1}^{2a_1}\cdot\ldots\cdot{p_n}^{2a_n}. $$