Question with this proof

Question

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 $m = qn + r$.

Answer 1

Suppose $m$ is odd; then $m$ is not even. What remainder does $m$ leave upon division by $2$?

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Suppose $m = 2q + 1$; conclude that $m$ is not even, since it leaves a non-zero remainder upon division by $2$. Thus, $m$ is odd.

Answer 2

$m\,$ is odd $\iff 2\nmid m\iff m\ {\rm mod}\ 2 \ne 0\iff m\ {\rm mod}\ 2 = 1\iff m = 2q+1$