The order of all elements of (Z/ 7161 Z)* divide 30

[Anoonumos]

Hi,

Homework Statement

Show that for every x in (Z/ 7161 Z)*, the order of x divides 30.

Homework Equations

(Z/ 7161 Z)* is the group of units of Z/ 7161 Z.

The Attempt at a Solution

I factorised 7161: 7161 = 3 * 7 * 11 * 31
I used the Chinese remainder theorem to show that (Z/ 7161 Z)* has (3-1)*(7-1)*(11-1)*(31-1) = 3600 elements.
So the order of every x in (Z/ 7161 Z)* has to divide 3600.
I don't know how to reduce this to 30. Can anyone help me with the next step?

Thanks.

 

[I like Serena]

It looks like you've applied Euler's theorem ($\phi$ function) instead of the Chinese remainder theorem.

Chinese remainder theorem says that if p,q,r relatively prime, that then:
$$(Z/pqrZ)^* \cong (Z/pZ)^* \times (Z/qZ)^* \times (Z/rZ)^*$$

Can you split your group like this?
And can you say something about the order of an element x in each of these groups?