From our lecture, I noted down this statement: Let $a$ be an integer and $m$ an odd natural number with the prime decomposition $$ m = p_1^{\alpha_1}\cdot p_2^{\alpha_2} \cdots p_n^{\alpha_n}. $$ If $a$ is a quadratic residue mod $m$, then $a$ is quadratic residue for every divisor of $m$.
We also have a proof and it is an easy one: Let $a$ be a quadratic residue mod $m$. Then we have an integer $x$ sch that $$ a\equiv x^2 \pmod m \Leftrightarrow m\mid a-x^2 \Rightarrow p\mid a-x^2 \Leftrightarrow a\equiv x^2 \pmod p $$ for every divisor $p$ of $m$.
But I am wondering why we did this for an odd $m$. It should also be possible for even numbers $m$, and the proof that we have does not distinguish. What am I missing?