Question 160
Let $p$, $q$ and $r$ be different prime numbers. For a positive integer $n$, let $f(n)$ denote the greatest common divisor of $n$, $p$, $q$ and $r$. Find the number of triples $(a, b, c)$ such that $1 \le a, b, c \le pqr$ and $f(a)$, $f(b)$, $f(c)$, $f(a+b)$, $f(b+c)$, $f(c+a)$, $f(a+b+c)$ are mutually different from each other.