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Multiplication Tables for the Group of Integers Modulo n

​
n
40
size
16
Given a positive integer
n
, the set
*
n
of positive integers coprime to
n
satisfies the axioms for an Abelian group under the operation of multiplication modulo
n
. For instance,
+
30
={1,7,11,13,17,19,23,29}
and
7*17=29
because
7×17=119≡29(mod30)
. This Demonstration shows the array plot of the multiplication table modulo
n
corresponding to
*
n
.

Details

The order of
*
n
is given by Euler's totient function
ϕ(n)
, implemented in Mathematica as EulerPhi[n], which for
n=0,1,2,3,…,16
has values
ϕ(n)=0,1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8
.
*
n
is cyclic only if
n
is
2,4,
k
p
, or
2
k
p
, where
p
is an odd prime and
k>0
. The first few values for which
*
n
is not cyclic are
8,12,15,16,20,21,24,28
. Any generator in the cyclic case is called a primitive root modulo
n
.

References

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

External Links

Permanent Citation

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