prove that if $m$ and $n$ are integers and $m+n$ is odd then $m-n$ is odd.

Question

prove that if m and n are integers and m+n is odd then m-n is odd.

Answer 1

$$m+n$$ is odd, this means $$m+n=2k+1$$ then we have $$m-n=2k-2n+1=2(k-n)+1$$ $$m+n-2n=2k+1-2n$$

Answer 2

Modulo $2,$ $-1\equiv1$, so $m+(-1)n\equiv m+n$

Answer 3

If $m$ and $n$ are both odd or both even, then $m+n$ (and $m-n$ as well) is even.

If $m$ is odd and $n$ is even, or the other way round, then $m+n$ is odd, but then $m-n$ is odd too. (And vice versa.)