SIMON-MARAIS 2017 A3

For each positive integer $n$, let $M(n)$ be the $n \times n$ matrix whose $(i,j)$ entry is equal to 1 if $i + 1$ is divisible by $j$, and equal to 0 otherwise.

Prove that $M(n)$ is invertible if and only if $n + 1$ is square-free.

(An integer is square-free if it is not divisible by the square of an integer larger than 1.)