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.

Re: Multiplication Modulo n on Z_m

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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