Is it necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? As the title suggests, is it a necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? Like, I understand the dictionary definitions of necessary and sufficient, but I don't have an idea on how to logically prove this problem...

The main result is Euclid's lemma:

If $p$ is prime and $p$ divides $ab$, then $p$ divides $a$ or $p$ divides $b$.

Now $6$ divides $m$ iff $2$ and $3$ divide $m$.
Apply these two facts to $6$ divides $n^2$ to conclude that $2$ and $3$ divide $n^2$ and so that $2$ and $3$ divide $n$.
