2n^2=m^2
Why is 2 a factor of m?
So, m and n are integers.
If m^2 is twice n^2 that means m^2 is even. (Which is the same as saying 2 is a factor of m^2.) But if m^2 is even then m must also be even.
As it turns out, we know something stronger: that 4 is a factor of m.
Are you looking at a proof that the square root of 2 is irrational, by chance?
if m is even say 2k for some integer k, then $\displaystyle m^2=(2k)^2=4k^2$ and so $\displaystyle m^2$ is even.
if m is odd say 2k+1 for some integer k, then $\displaystyle m^2=(2k+1)^2=4k^2+4k+1$ is odd.
so if 2 divides $\displaystyle m^2$, then m^2 is even and by above m is even and so 2 divides m.