11. $[\mathbf{8}]$ For positive integers $m, n$, let $\operatorname{gcd}(m, n)$ denote the largest positive integer that is a factor of both $m$ and $n$. Compute
$$
\sum_{n=1}^{91} \operatorname{gcd}(n, 91)
$$
Answer: 325 Since $91=7 \times 13$, we see that the possible values of $\operatorname{gcd}(n, 91)$ are $1,7,13,91$. For $1 \leq n \leq 91$, there is only one value of $n$ such that $\operatorname{gcd}(n, 91)=91$. Then, we see that there are 12 values of $n$ for which $\operatorname{gcd}(n, 91)=7$ (namely, multiples of 7 other than 91 ), 6 values of $n$ for which $\operatorname{gcd}(n, 91)=13$ (the multiples of 13 other than 91 ), and $91-1-6-12=72$ values of $n$ for which $\operatorname{gcd}(n, 91)=1$. Hence, our answer is $1 \times 91+12 \times 7+6 \times 13+72 \times 1=325$.