I need to prove that if both $ m$ and $n$ are odd integers, then $mn+2$ is odd. I found several similar answers for a problem that asks if $m$ and $n$ are odd the prove that $mn$ is odd as well. This question adds an even number and wants to prove that it is odd.
If you know that $m$ is odd then you can say that it is in some way equal to $(2p+1)$. Likewise, $n$ is also equal to $(2p+1)$. Then multiply these together to get $mn=4p^2+4p+1$. This must be odd because we can set $4p^2+4p+1=2(2p^2+2p)+1$ and if $(2p^2+2p)$ is set equal to $b$ then $mn=2b+1$ which is an odd number. Adding $2$ does not change our answer because if $mn=4p^2+4p+1$ and we add $2$, $mn+2=4p^2+4p+1+2$, then we can say that if $g=(2p^2+2p+1)$ then $mn+2=2(2p^2+2p+1)+1=2g+1$ which is odd.