Euler's Totient Function Problem with my Understanding

Question

Solve φ36

List of relatively prime numbers: 1,2,3,4,6,9,12,18,36

So from this I would say that φ36 = 9, but using an online calculator I got it to be 12. Where am I going wrong?

Answer 1

The Euler Totient function is multiplicative, i.e., $\varphi(1)=1$ and for comprime $m,n$, $\varphi(mn)=\varphi(m)\varphi(n)$.

Using this, we have $\varphi(36)=\varphi(4)\varphi(9)=2*6=12$.

Note that $\varphi(4)=2$ because the numbers relatively prime to $4$ are $1$ and $3$; $\varphi(9)=6$ since the numbers relatively prime to $9$ are $1,2,4,5,7,8$.

Answer 2

Your list of relatively prime numbers is in reality a list of factors. A number relatively prime to $36$ must have no factors of $2$ or $3$, like $5, 7, 11, 35$