ISL 2014 N6

Let $a_{1}<a_{2}<\cdots<a_{n}$ be pairwise coprime positive integers with $a_{1}$ being prime and $a_{1} \geqslant n+2$. On the segment $I=\left[0, a_{1} a_{2} \cdots a_{n}\right]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_{1}, \ldots, a_{n}$. These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_{1}$. (Serbia)