Interpreting Euler phi function using probability

Question

I am trying to figure out how to interpret the following formula using probability:

$$\frac{\phi(n)}{n}=\prod_{p|n}(1-\frac{1}{p}).$$

The left hand side is clearly the probability that a random chosed number from $1\leq a\leq n$ is coprime to $n$.

I am told that the right hand side insists that our number is not divisible by any prime divisors of $n$. Why is this so?

Answer 1

Because of $1\le \phi(n)\le n$ we have $0<\frac{\phi(n)}{n}\le 1$. Hence the RHS satisfies $$ 0<\prod_{p|n}(1-\frac{1}{p})\le 1. $$ So the claim about divisibility by primes is clear.

Answer 2

Coprime means, not sharing any factor other than 1 in common. Therefore, to count the number of coprimes less than N, is exactly equal to, N times the product of 1 minus 1 pth, where p ranges over the prime factors of N. A number in general, is coprime to N, if it is in one of that many remainders mod N.