# Math Help - Phi Function

1. ## Phi Function

Let & denote the phi function

Show that if m and n are not relatively prime, then &(m)&(n) < &(mn)

2. Well, if $(m,n)>1$ then they must have a common prime divisor $p$.

Let $p^{\alpha}$ and $p^{\beta}$ be the greatest powers of p dividng $m$ and $n$ respectively, then $p^{\alpha+\beta}$ is the greatest power of p dividng $m\cdot n$.

But then $\phi\left(p^{\alpha}\right)\cdot \phi\left(p^{\beta}\right) < \phi\left(p^{\alpha+\beta}\right)$ and you can finish it off by remembering $\phi$ is multiplicative.