7.
a. Identify the error in the following "proof."
Let $m$, $n$ be two integers. If $m$ is even and gcd($m$, $n$) = 1, then $n$ is odd.
If gcd($m$, $n$) = 1, then 1 = $mx + ny$ and $m = 2y$ for some integers $x$, $y$. Hence,
1 = ($2y$)x + $ny$ = $y$($2x + n$). So, $2x + n$ | 1 and $2x + n = \pm 1$, and in either case $n = 2(-x) \pm 1$,
and therefore $n$ is odd.
b. Determine whether the conditional in part (a) is true or false. If it is true, give a correct proof; if it
is not true, give a counterexample.
