CN-TST 2014 Problem 18

For positive integer $k>1$ , let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$ , as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$ . Prove: If $n$ is a positive integer greater than $1$ , $p$ is a prime factor of $n$ , then $f(n)\leq \frac{n}{p}$