Sums of squares of positive integers prime to n

Let denote the sum of the squares of the positive integers and prime to .
Prove that


I have trouble with separating the integers into classes, so that all integers such that are in the same class.
Thanks for the help.

Let , for each d|n, define , then is a partion on N.
And for each , Since , By the definition of S(.), the sum of elements in is ,
Thus,the first equality of
is proved.
When d exhaust all the divisors of n, so do , Thus the second equality obviously hold.