# prove that if $m$ and $n$ are odd integers, then $mn+2$ is odd.

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.

• Welcome to Mathematics Stack Exchange. Odd + Even = Odd – J. W. Tanner Oct 27 '20 at 2:05
• Break it into steps: first show that $mn$ is odd, then that $mn+2$ is also. – abiessu Oct 27 '20 at 2:07

## 3 Answers

If $$m=2k+1$$ and $$n=2l+1$$ for some integers $$k$$ and $$l$$,

then $$mn+2=4kl+2k+2l+3=2(2kl+k+l+1)+1$$.

• Thank you very much. – sinco demayo Oct 27 '20 at 2:18

If $$m$$ and $$n$$ are odes then $$m+1, n+1$$ and $$m+n$$ are even.

In particular, $$(m+1)(n+1)-(m+n)$$ is even. But $$mn+2=(m+1)(n+1)-(m+n)+1$$ is odd.

• Thank you euler88. – sinco demayo Oct 27 '20 at 2:19

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.