0
$\begingroup$

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.

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

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

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

$\endgroup$
1
  • 1
    $\begingroup$ Thank you very much. $\endgroup$ – sinco demayo Oct 27 '20 at 2:18
0
$\begingroup$

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.

$\endgroup$
1
  • 1
    $\begingroup$ Thank you euler88. $\endgroup$ – sinco demayo Oct 27 '20 at 2:19
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.