Inverting arithmetic functions

Question

I know that if $$F(n)=\sum_{d|n}f(d)$$ then $$f(p^e)=F(p^e)-F(p^{e-1})$$ Let $F(n)=\sum_{d|n}f(d)$ and assume $F(n)$ is multiplicative, so $F(1)=1$. Find the formula for $f(n)$ when $F(n)=1$ for all $n$. Because $F(n)=1$ for all $n$, we can write $n=p^e$ By plugging $F(p^e)=1$ into the formula, I get $$f(p^e)=1-1=0$$ but this doesn't seem correct. Do I need to set $F(1)=f(1)=1$ and then let every other term $f(n)=0$ to satisfy the sum?