RO-NMO 2015 Problem 102

Consider a natural number $n$ for which it exist a natural number $k$ and $k$ distinct primes so that $n=p_1\cdot p_2\cdots p_k.$ a) Find the number of functions $f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\}$ that have the property that $f(1)\cdot f(2)\cdots f\left( n \right)$ divides $n.$ b) If $n=6,$ find the number of functions $f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\}$ that have the property that $f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6)$ divides $36.$