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NumberTheory

 MultiplicativeOrder
 order of a number under modular multiplication

 Calling Sequence MultiplicativeOrder(m, n)

Parameters

 m - positive integer n - positive integer greater than 1

Description

 • The MultiplicativeOrder function computes the multiplicative order of m modulo n, which is defined as the least positive integer exponent i such that m^i is congruent to $1$ modulo n.
 • Alternatively, the multiplicative order can be defined as the order of the cyclic group generated by m under multiplication modulo n.
 • The multiplicative order exists if and only if m and n are coprime. In the case that it does not exist, an error message is displayed.

Examples

 > with(NumberTheory):
 > MultiplicativeOrder(7, 18);
 ${3}$ (1)
 > seq(7^i mod 18, i = 1..3);
 ${7}{,}{13}{,}{1}$ (2)

If the multiplicative order of m is equal to the totient of n, then m is a primitive root modulo n.

 > Totient(18);
 ${6}$ (3)
 > MultiplicativeOrder(11, 18);
 ${6}$ (4)
 > PrimitiveRoot(18, greaterthan = 10);
 ${11}$ (5)

Since $5$ and $25$ are not coprime, the multiplicative order of $5$ modulo $25$ is not defined.

 > MultiplicativeOrder(5, 25);

Compatibility

 • The NumberTheory[MultiplicativeOrder] command was introduced in Maple 2016.