RO-TST 2012 Problem 12

Let $a_1$ , $\ldots$ , $a_n$ be positive integers and $a$ a positive integer that is greater than $1$ and is divisible by the product $a_1a_2\ldots a_n$ . Prove that $a^{n+1}+a-1$ is not divisible by the product $(a+a_1-1)(a+a_2-1)\ldots(a+a_n-1)$ .