# Multiplication Modulo n on Z_m

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• Oct 15th 2012, 05:53 AM
ncshields
Multiplication Modulo n on Z_m
We showed that Z_m - {0} = {1,2,...,m-1} is not always a group under multiplication modulo m. Write Z*_m for the set of all elements in Z_m which have a multiplicative inverse in Z_m.
(a) Prove that a in Z*_m if and only if (a; m) = 1. Conclude that Z*_m has exactly phi(m) elements.
(b) Verify that Z*_m is a group under multiplication modulo m. Conclude that a phi(m)= 1 for a in Z*_m.
• Oct 15th 2012, 06:39 AM
topsquark
Re: Multiplication Modulo n on Z_m
Quote:

Originally Posted by ncshields
We showed that Z_m - {0} = {1,2,...,m-1} is not always a group under multiplication modulo m. Write Z*_m for the set of all elements in Z_m which have a multiplicative inverse in Z_m.
(a) Prove that a in Z*_m if and only if (a; m) = 1. Conclude that Z*_m has exactly phi(m) elements.
(b) Verify that Z*_m is a group under multiplication modulo m. Conclude that a phi(m)= 1 for a in Z*_m.

What have your tried so far?

-Dan