How to prove $\frac{(a_1 a_2\cdots a_n)^2-1}{8}\equiv\sum_{i=1}^n\frac{a^2_i -1}{8}\pmod 8$ Let $a_1,a_2,\cdots,a_n$ be odd numbers, show that
$$\frac{(a_{1}a_{2}\cdots a_{n})^2-1}{8}\equiv\sum_{i=1}^{n}\dfrac{a^2_{i}-1}{8} \pmod 8$$
Special cases
$n=1$: It is obvious that
$$\frac{a^2_{1}-1}{8}\equiv\dfrac{a^2_{1}-1}{8}\pmod 8$$
$n=2$
$$\dfrac{a^2_{1}a^2_{2}-1}{8}\equiv\dfrac{(2k-1)^2(2m-1)^2-1}{8}\equiv\frac{4k^2-4k+4m^2-4m}{8}=\dfrac{a^2_{1}+a^2_{2}-2}{8}\pmod 8?$$
where $a_{1}=2k-1,a_{2}=2m-1$
because
$$(2k-1)(2m-1)^2-1=(4km-2k-2m+1)^2-1=(16k^2m^2-16k^2m-16km^2+8km+8km)+4k^2+4m^2-4k-4m+1-1$$
But in general I can't prove it.
 A: We prove the congruence by induction on the number of odd factors.
For $n = 1$, the congruence is even an equality, so the base case is settled.
Now assume that $n > 1$ and, as the induction hypothesis, that the congruence
$$\frac{\left(\prod_{k=1}^{n-1}a_k\right)^2 - 1}{8} \equiv \left(\sum_{k=1}^{n-1} \frac{a_k^2-1}{8}\right) \pmod{8}$$
holds for products of $n-1$ odd integers $a_k$.
The induction step then goes:
\begin{align}
\frac{\left(\prod_{k=1}^n a_k\right)^2-1}{8} - \frac{a_n^2-1}{8}
&= \frac{\left(\prod_{k=1}^n a_k\right)^2 - a_n^2}{8}\\
&= a_n^2\frac{\left(\prod_{k=1}^{n-1} a_k\right)^2-1}{8}\\
&\equiv \left(a_n^2\sum_{k=1}^{n-1} \frac{a_k^2-1}{8}\right) \pmod{8}\tag{IH}\\
&\equiv \left(\sum_{k=1}^{n-1} \frac{a_k^2-1}{8}\right) \pmod{8}, \tag{S}
\end{align}
where IH denotes the induction hypothesis, and S the congruence $a_n^2\equiv 1 \pmod{8}$ holding for all odd integers. Rearranging yields the congruence in the desired form
$$\frac{\left(\prod_{k=1}^n a_k\right)^2-1}{8} \equiv \left(\sum_{k=1}^n \frac{a_k^2-1}{8}\right) \pmod{8}.$$
