Let $a_1, a_2,\cdots ,a_n$ be positive integers and let $a$ be an integer greater than $1$ and divisible by the product $a_1a_2\cdots a_n$. Prove that $a^{n+1} + a-1$ is not divisible by the product $(a + a_1 - 1)(a + a_2 - 1) \cdots (a + a_n - 1)$.
Source: Indian Postal Coaching 2012 Set 1 #2
Tags: number theory unsolved, number theory
Let $a_1, a_2,\cdots ,a_n$ be positive integers and let $a$ be an integer greater than $1$ and divisible by the product $a_1a_2\cdots a_n$. Prove that $a^{n+1} + a-1$ is not divisible by the product $(a + a_1 - 1)(a + a_2 - 1) \cdots (a + a_n - 1)$.