KöMaL 2017 B4872

Let $p$, $q$ and $r$ denote three distinct primes, and let $n=pq^2r^3$. Show that $(n,1)+(n,2)+\dots+(n,n) =qr^2(2p-1)(3q-2)(4r-3),$ where $(a,b)$ denotes the greatest common divisor of $a$ and $b$.