Question 42
For a positive integer $n$, let $\varphi(n)$ denote the number of positive integers less than and relatively prime to $n$. Let $S_{k}=\sum_{n} \frac{\varphi(n)}{n}$, where $n$ runs through all positive divisors of $42^{k}$. Find the largest positive integer $k \lt 1000$ such that $S_{k}$ is an integer.