Question 54
Let $n$ be the product of the first 10 primes, and let
$$
S=\sum_{x y \mid n} \varphi(x) \cdot y
$$
where $\varphi(x)$ denotes the number of positive integers less than or equal to $x$ that are relatively prime to $x$, and the sum is taken over ordered pairs $(x, y)$ of positive integers for which $x y$ divides $n$. Compute $\frac{S}{n}$.
$$
S=\sum_{x y \mid n} \varphi(x) \cdot y
$$
where $\varphi(x)$ denotes the number of positive integers less than or equal to $x$ that are relatively prime to $x$, and the sum is taken over ordered pairs $(x, y)$ of positive integers for which $x y$ divides $n$. Compute $\frac{S}{n}$.