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Multiplication Tables for the Group of Integers Modulo n

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 .

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The order of is given by Euler's totient function , implemented in Mathematica as EulerPhi[n], which for has values . is cyclic only if is , 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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