How to compute the totient function when the input is not a prime?

Question

I've been searching on google how to solve this question but I only understand how to solve when it's a prime, which should be $\phi(p)=p-1$ for prime $p$ if I've understand correctly.

But the question is "Find the Euler totient function of $272$."

$272$ is not a prime, so how would I solve this?

Welcome to MSE. Do you know the definition of the totient function? It's defined for all natural numbers, not just for primes. — José Carlos Santos, Feb 15 '23 at 10:22
If you know the prime decomposition of $n$ then you can easily compute $\varphi(n)$. If you don't know the formula then it's strange you were given this exercise. — Mark, Feb 15 '23 at 10:25

score 1 · Accepted Answer · answered Feb 15 '23 at 10:33

1

Use the fact that the totient function is multiplicative so for coprime $m,n$ $\phi(m\times n)=\phi(m)\times\phi(n)$ and that for primes $p$, $\phi(p^k)=p^{k-1}(p-1)$. Now decompose $272$ into prime powers.

answered Feb 15 '23 at 10:33

Vivaan Daga

6,072

How to compute the totient function when the input is not a prime?

1 Answers1