Question 29
For positive integer $k \gt 1$, 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 counted, for example $f(12) = 4$, as $12$ can be factored in these 4 ways: $12$, $2 \times 6$, $3 \times 4$, $2 \times 2 \times 3$).
**Prove:** If $n$ is a positive integer greater than 1, $p$ is a prime factor of $n$, then $f(n) \le \frac{n}{p}$.
**Prove:** If $n$ is a positive integer greater than 1, $p$ is a prime factor of $n$, then $f(n) \le \frac{n}{p}$.