Proof that $n$ is a positive odd integer if $5n + 6$ is odd

Question

Prove that $n$ is a positive odd integer if

$5n + 6 = 2k + 1$ or $(5n + 6) = 1\mod 2$

Answer 1

This statement is wrong! Try $n=\frac{1}{5}.$

Answer 2

What you want to prove is not true -- it fails, for example for $n=-1$.

$5\cdot(-1)+6 = 1$ which is odd, but $-1$ is not a positive odd integer.