 Solution

The inclusion-exclusion principle says that the size of a finite union is the alternating sum of the sizes of all nonempty intersections of its constituent sets.

For each prime number $p\mid n$, let $A_p$ consist of the tuples for which $p$ divides every $x_i$. Exactly $(n/p{)}^r$ tuples lie in $A_p$, and for distinct primes $p_1,\dots ,p_j\mid n$, exactly $(n/(p_1\cdots p_j){)}^r$ lie in their intersection. A tuple has greatest common divisor greater than one with $n$ exactly when it belongs to some $A_p$. Inclusion-exclusion therefore gives the Jordan totient function