Show $8\mid n^2-1$ if $n$ is an odd positive integer. Show that $n^2-1$ is divisible by $8$, if $n$ is an odd positive integer.
Please help me to prove whether this statement is true or false.

Viewing the integer numbers modulo 8, write $a \equiv b$ for $8|(a-b)$.
(This structure, $\mathbb Z_8$ is generated by $8\equiv 0$ and is friendly with operations $+$ and $\cdot$.)
We have the following set of odd numbers: $\{ 1,3,5,7\}$. Or, rewriting by $5\equiv-3$ and $7\equiv -1$, this is only
$$\{ 1,3,-3,-1\}$$
The squares of these:
$$(\pm 1)^2 = 1\ \text{ and }\ (\pm 3)^2=9\equiv 1$$
