11. Let $f(n)$ be a function defined on the set of positive integers (called a number-theoretic function), and let $F(n)=\sum_{d \mid n} f(d)$. Prove: If for any $\left(n_{1}, n_{2}\right)=1$ there must be $f\left(n_{1} n_{2}\right)=$ $f\left(n_{1}\right) f\left(n_{2}\right)$, then for any $\left(n_{1}, n_{2}\right)=1$ there must also be
$$F\left(n_{1} n_{2}\right)=F\left(n_{1}\right) F\left(n_{2}\right)$$