TITLE: Question with this proof
QUESTION [2 upvotes]: The integer $m$ is odd if and only if there exists q $\in \mathbb{Z}$ such that $m=2q+1$
I know that $m$ is even if 2|n, and $n$ is odd if $n$ is not even. I also know the division algorithm, which is that for every $m$ there exists $m = qn + r$.

REPLY [1 votes]: $m\,$ is odd $\iff 2\nmid m\iff m\ {\rm mod}\ 2 \ne 0\iff m\ {\rm mod}\ 2 = 1\iff m = 2q+1$
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