Wolfram Demonstrations Project

Given a positive integer

, the set

of positive integers coprime to

satisfies the axioms for an Abelian group under the operation of multiplication modulo

. For instance,

and

because

. This Demonstration shows the array plot of the multiplication table modulo

corresponding to

The order of

is given by Euler's totient function

, implemented in Mathematica as EulerPhi[n], which for

has values

is cyclic only if

, or

, where

is an odd prime and

. The first few values for which

is not cyclic are

. Any generator in the cyclic case is called a primitive root modulo

Reference

[1] Wikipedia. "Multiplicative Group of Integers Modulo n." (Jul 31, 2012) en.wikipedia.org/wiki/Multiplicative_group_of _integers _modulo _n.

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Multiplication Tables for the Group of Integers Modulo n