# Is it possible to prove directly that even perfect squares have even square roots? Or, symbolically: forall n in Z, n^2 is even =>n is even

Diana Suarez 2022-09-27 Answered
Is it possible to prove directly that even perfect squares have even square roots? Or, symbolically:
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GrEettarim3
Since $2$ is a prime, and since $p|ab$ implies either $p|a$ or $p|b$, that tells you that $2|{n}^{2}$ implies that $2|n$.
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Ryland Houston
Let $n={m}^{2}$. Let $m=\prod {p}_{i}^{{n}_{i}}$ be the unique prime factorization of $m$. Then ${n}^{2}=\prod {p}_{i}^{2{n}_{i}}$ is the prime factorization of ${n}^{2}$. As ${n}^{2}$ is even one of those primes is $2$. Those are the primes that factor $m$ so $m$ is even.