Euler totient function

Definitions
The Euler-phi function on a natural number $$n$$, denoted $$\varphi(n)$$, is defined in the following equivalent ways : 1. as the number of positive integers not greater than $$n$$ that are coprime to $$n$$. 2. as the number of generators (cyclic elements) in a cyclic group of order $$n$$.

Every natural number is the sum of Euler-phi function on positive divisors
For any natural number $$n$$, we have $$n=\sum_{d|n}\varphi(n)$$, where the summation runs over all positive divisors of $$n$$.

Evaluation on prime powers
For any prime $$p$$ and positive integer $$k$$, the value of Euler-phi function on $$p^{k}$$ is given by the formula $$\varphi(p^{k})=p^{k-1}(p-1)$$. This is an immediate application of the previous fact.

Multiplicative if coprime
The Euler-phi function is multiplicative on coprime numbers, that is, if $$m$$ and $$n$$ are coprime, then $$\varphi(mn)=\varphi(m)\varphi(n)$$. For full proof, see Euler-phi function is multiplicative if coprime.

Explicit formula
Knowing the prime factorization of $$n$$, we can evaluate $$\varphi(n)$$ by repeated applications of the above facts. However, knowing only the distinct prime factors of the number is sufficient, because the value of Euler-phi function on it is then given by the formula $$\varphi(n)=n\prod_{p}(1-\frac1{p})$$, where here the product is evaluated over all distinct prime factors of $$n$$.