Prime Factors Problem

The number of relatively prime or coprime positive integers less than $n$ is given by the Euler's totient function $\phi (n)$ . If $n = p_1^{q_1} p_2^{q_2} p_3^{q_3} \cdots p_m^{q_m}$ , where $p_1, p_2, p_3, \cdots p_m$ are the prime factors of $n$ and $q_1, q_2, q_3, \cdots q_m$ are their respective powers, then

$\phi(n) = n \left(1 - \frac 1{p_1} \right) \left(1 - \frac 1{p_2} \right) \left(1 - \frac 1{p_3} \right) \cdots \left(1 - \frac 1{p_m} \right)$

Therefore,

$\begin{aligned} \phi (180) & = 180 \left(1 - \frac 12 \right) \left(1 - \frac 13 \right) \left(1 - \frac 15 \right) \\ & = 180 \left(\frac 12 \right) \left(\frac 23 \right) \left(\frac 45 \right) \\ & = \boxed {48} \end{aligned}$

It's all about Euler's totient function. $\phi (180)=\phi (2^23^25^1)=2^{2-1}(2-1)\times 3^{2-1}(3-1)\times (5-1)=\boxed {48}$ .