# Multiplicative order

In number theory, given an integer a and a positive integer n with gcd(a,n) = 1, the multiplicative order of a modulo n is the smallest positive integer k with

${\displaystyle a^{k}\ \equiv \ 1{\pmod {n}}\,}$

In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n.

The order of a modulo n is usually written as:

${\displaystyle k=\operatorname {ord} _{n}(a)}$

or

${\displaystyle k=O_{n}(a)}$

## Example

The powers of 4 modulo 7 are as follows:

${\displaystyle {\begin{array}{llll}4^{0}&=1&=0\times 7+1&\equiv 1{\pmod {7}}\\4^{1}&=4&=0\times 7+4&\equiv 4{\pmod {7}}\\4^{2}&=16&=2\times 7+2&\equiv 2{\pmod {7}}\\4^{3}&=64&=9\times 7+1&\equiv 1{\pmod {7}}\\4^{4}&=256&=36\times 7+4&\equiv 4{\pmod {7}}\\4^{5}&=1024&=146\times 7+2&\equiv 2{\pmod {7}}\\\end{array}}}$
${\displaystyle {\text{...etc...}}}$

The smallest positive integer k such that 4k = 1 (mod 7) is 3, so O7(4) = 3.

## Properties

Even without knowledge that we are working in the multiplicative group of integers modulo n, we can show that a actually has an order by noting that the powers of a can only take a finite number of different values modulo n, so according to the pigeonhole principle there must be two powers, say s and t and without loss of generality s > t, such that as ≡ at (mod n). Since a and n are coprime, this implies that a has an inverse element a−1 and we can multiply both sides of the congruence with at, yielding ast ≡ 1 (mod n).

The concept of multiplicative order is a special case of the order of group elements. The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Zn; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or U(Zn).

As a consequence of Lagrange's theorem, ordn(a) always divides φ(n). If ordn(a) is actually equal to φ(n) and therefore as large as possible, then a is called a primitive root modulo n. This means that the group U(n) is cyclic and the residue class of a generates it.

The order ordn a also divides λ(n), a value of the Carmichael function, which is an even stronger statement than the divisibility of φ(n).