Simple proof involving integers

Question

Does this look ok to you?

let $m, n$ be any integers and $mn$ and $m + n$ are both even, prove that $m$ and $n$ are both even.

So $mn$ and $m+n$ are integers from what we are given we can assume even integers.

$mn+m+n=2(j+p)$ for some $j$ and $p$ in the integers

$(m+1)(n+1)=2(j+p)+1$

so $(m+1)(n+1)$ is odd it follows $m+1$ and $n+1$ are odd and so $n$ and $m$ are even.

Thanks a lot guys

Yes, that is fine, assuming that you have already proven in an in-class example or homework etc... that the product of two integers is odd if and only if each of the two integers is odd. If that has not been established yet, then it too must be proven. — JMoravitz, Nov 07 '18 at 00:43
Oh i dont think we have, i might have to go back to the drawing board — hitherematey, Nov 07 '18 at 00:46

score 1 · Accepted Answer · answered Nov 07 '18 at 00:50

1

Yes your proof is correct.

What I really like about it is avoiding different cases for $m$ and $n$

You proved your statement in one shot and that deserves merit.

answered Nov 07 '18 at 00:50

1 Answers1