# Thread: a few questions on (Z/nZ)* and Euler's Totient Function

1. ## a few questions on (Z/nZ)* and Euler's Totient Function

phi(n) Euler's Totient Function:

Given the group (Z/nZ)* would I be correct in saying that L = lcm(d1,...dk) where d1,...,dk are the distinct orders of the elements of (Z/nZ)* is the smallest integer for which a^L=1 for all elements of (Z/nZ)*?

Secondly, since di | |(Z/nZ)*| = phi(n) for i=1,...,k am I correct in saying that L is less than or equal to phi(n)?

Main question(s):

When is phi(n)/L an integer (always?)? More generally, can we write phi(n)/L as f(phi(n)) some function of phi(n)? When is phi(n) an element of (Z/nZ)*?

Cheers. Any help (on any of the questions) much appreciated.

2. ## Re: a few questions on (Z/nZ)* and Euler's Totient Function

Originally Posted by b5345007
phi(n) Euler's Totient Function:

Given the group (Z/nZ)* would I be correct in saying that L = lcm(d1,...dk) where d1,...,dk are the distinct orders of the elements of (Z/nZ)* is the smallest integer for which a^L=1 for all elements of (Z/nZ)*?
Yes.

Secondly, since di | |(Z/nZ)*| = phi(n) for i=1,...,k am I correct in saying that L is less than or equal to phi(n)?
Yes.

Main question(s):

When is phi(n)/L an integer (always?)?
Yes, always.

More generally, can we write phi(n)/L as f(phi(n)) some function of phi(n)?
No. If we could then we would have $2=\varphi(8)/L_8=f(\varphi(8))=f(4)=f(\varphi(5))=\varphi(5)/L_5=1$, a contradiction.

When is phi(n) an element of (Z/nZ)*?
$\varphi(n)$ is an element of $(\mathbb{Z}/n\mathbb{Z})^\times$ if and only if all of the following are false:

(1) $n=2r$ for $r>2$;

(2) $p^2\big|n$ for some prime $p$;

(3) $p$ divides $(q-1)/2$ for some odd prime divisors $p,q$ of $n$.

Cheers. Any help (on any of the questions) much appreciated.

Cheers!