1) Show that ifpis prime, $\displaystyle 2^ap+1$ is composite fora= 1,2,....,r andpis not a Fermat prime, whereris a positive integer, then $\displaystyle \phi(n) = 2^rp$ has no solution.

2) The arithmetic funtiongis said to be theinverseof the arithmetic functionfiff * g = g * f = i. Show that the arithmetic functionfhas aninverseif and only if f(1) does not equal 0. Show that iffhas an inverse it is unique.

(Hint: When f(1) is not equal to 0, find the inverse $\displaystyle f^{-1}$ offby calculating $\displaystyle f^{-1}(n)$ recursively, using the fact that i(n) = summation $\displaystyle (f(d)f^{-1}(n/d)$.)