Multiplicative Functions Proof

Question

I don't know if this question was asked before, but I could not find it on this forum: a function $f(n)$ is multiplicative if $f(mn)=f(m)f(n)$, where m and n are coprime and positive. If $d(n)$ is the number of positive divisors of n, how can I prove $d(n)$ is multiplicative? Thanks.

Answer 1

Welcome to Math SX!

Note that if two numbers are coprime, they do not have common divisors (except for $1$), therefore every combination $pq$ (where $p$ is divisor of $m$ and $q$ is a divisor of $n$) is a divisor of $mn$. Using this, the result will follow (I hope you can see it).