HMMT November 2024 U27

For any positive integer $n$, let $f(n)$ be the number of ordered triples $(a, b, c)$ of positive integers such that

• max $(a, b, c)$ divides $n$ and
• $\operatorname{gcd}(a, b, c)=1$.

Compute $f(1)+f(2)+\cdots+f(100)$.