Lemma (3.1)

Let $G$ be a cyclic group of order $n$. Then, for any divisor $d$, $G$ contains an element of order $d$.

Lemma (3.2)

If $a$ is odd, then $8$ divides $a^{2}-1$. For $a,b$ odd, We have $$\frac{a^{2}b^{2}-1}{8}\equiv \frac{a^{2}-1}{8}+\frac{b^{2}-1}{8}\pmod{8}$$