Školjka

Let $a_1 < a_2 < \cdots < a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \geq n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n]$ 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 swquares of the length of these segments is divisible by $a_1$. \begin{flushright}\emph{(Serbia)}\end{flushright}