Given distinct positive integers $a_1,\,a_2,\,\dots,a_n$. Let $b_i = (a_i - a_1) (a_i-a_2) \dots (a_i-a_{i-1}) (a_i-a_{i+1})\dots(a_i-a_n)$. Prove that the least common multiple $[b_1,b_2,\dots,b_n]$ is divisible by $(n-1)!.$
Source: 239 2004 J P6
Tags: number theory, least common multiple
Given distinct positive integers $a_1,\,a_2,\,\dots,a_n$. Let $b_i = (a_i - a_1) (a_i-a_2) \dots (a_i-a_{i-1}) (a_i-a_{i+1})\dots(a_i-a_n)$. Prove that the least common multiple $[b_1,b_2,\dots,b_n]$ is divisible by $(n-1)!.$