Prove using the principle of inclusion and exclusion:
Let $n$ be a number that can be written as a product of prime numbers, $n=p_1^{i_1}p_2^{i_2}\cdots p_t^{i_t}$
Prove that the number of integers between in $[1,n]$ that are strangers (relatively prime) to $n$ is $n\prod_{i = 1}^{t}\left(1- \frac{1}{p_i}\right)$
I see that the group of element I am working with is all the integer number between $1$ and $n$. And I also know that to find the number of numbers between $1$ and $n$ that are divisible by $p$ a prime numbers I show divide $\left \lfloor{\frac{n}{p}}\right \rfloor $.
I got $\sum_{k=0}^{t} (-1)^k\left \lfloor{\frac{n}{p_k}}\right \rfloor=\sum_{k=0}^{t} (-1)^k\left \lfloor{\frac{p_1^{i_1}p_2^{i_2}...p_t^{i_t}}{p_k}}\right \rfloor. $
Thanks in advance!