Let $p$ be an odd prime, and $a_1,a_2,\ldots,a_n$ be a series of integers in arithmetic progression whose common difference is not divisible by $p$. Prove that there exists an index $i$ such that $\displaystyle\prod_{k=1}^p a_k+a_i$ is divisible by $p^2$.
Since $a_1,a_2,\ldots,a_n$ are in arithmetic progression with say common difference $d$, we have the sequence to be $a_1,a_1+d,a_1+2d,\ldots,a_1+(n-1)d$. We then have $$\displaystyle\prod_{k=1}^p a_k+a_i = a_1(a_1+d) \cdots (a_1+(n-1)d)+a_i.$$ How do we continue?