Let $n\in \mathbb{N}$. Prove that $lcm [1,2,..,n]=lcm [\binom{n}{1},\binom{n}{2},...,\binom{n}{n}]$ if and only if $n+1$ is a prime number.
Problem
Source: IV International Festival of Young Mathematicians Sozopol 2013, Theme for 10-12 grade
Tags: number theory, Lucas' Theorem