IT-CESENATICO 1998 Problem 6

We say that a function $f : N \to N$ is increasing if $f(n) < f(m)$ whenever $n < m$ , multiplicative if $f(nm) = f(n)f(m)$ whenever $n$ and $m$ are coprime, and completely multiplicative if $f(nm) = f(n)f(m)$ for all $n,m$ . (a) Prove that if $f$ is increasing then $f(n) \ge n$ for each $n$ . (b) Prove that if $f$ is increasing and completely multiplicative and $f(2) = 2$ , then $f(n) = n$ for all $n$ . (c) Does (b) remain true if the word "completely" is omitted?