Multiplication Tables for the Group of Integers Modulo n
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, ={1,7,11,13,17,19,23,29} and because . This Demonstration shows the array plot of the multiplication table modulo corresponding to .
n
*
n
n
n
+
30
7*17=29
7×17=119≡29(mod30)
n
*
n
Details
Details
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 .
*
n
ϕ(n)
n=0,1,2,3,…,16
ϕ(n)=0,1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8
*
n
n
2,4,
k
p
2
k
p
p
k>0
*
n
8,12,15,16,20,21,24,28
n
References
References
[1] Wikipedia. "Multiplicative Group of Integers Modulo n." (Jul 31, 2012) en.wikipedia.org/wiki/Multiplicative_group_of _integers _modulo _n.
External Links
External Links
Permanent Citation
Permanent Citation
Jaime Rangel-Mondragon
"Multiplication Tables for the Group of Integers Modulo n"
http://demonstrations.wolfram.com/MultiplicationTablesForTheGroupOfIntegersModuloN/
Wolfram Demonstrations Project
Published: August 2, 2012