Problem
ALG-B3-M05-P005 Multiplicativity on Natural Numbers
#5
★★☆☆☆ Level 2 of 5
Let \(f:\mathbb N\to\mathbb N\), \(f(mn)=f(m)f(n)\), and suppose that \(f(p)=p^2\) for every prime \(p\). Find \(f(n)\).
Factor \(n\) into primes.
If \(n=p_1^{a_1}\cdots p_k^{a_k}\), then by multiplicativity \(f(n)=f(p_1)^{a_1}\cdots f(p_k)^{a_k}=p_1^{2a_1}\cdots p_k^{2a_k}=n^2\). The answer is \(f(n)=n^2\).
Here the domain is \(\mathbb N\), so factorisation is the tool.