---
tag: 0Y4B7P0GS8
kind: statement
role: corollary
name: The order modulo p divides p − 1
msc: [11A07]
uses: [0SBRX5CZ5R, 0M25PRN3WQ]
sources:
  - url: https://en.wikipedia.org/wiki/Multiplicative_order
    locator: Properties
---

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]], and let $a$ be an integer with $p \nmid a$.
Then the [[multiplicative order|0SBRX5CZ5R]] of $a$ modulo $p$ divides $p - 1$.

## Remarks

The proof: $a^{p-1} \equiv 1 \pmod p$ by [[Fermat's little theorem|0M25PRN3WQ]], and each exponent $m$ with $a^m \equiv 1$ is a multiple of the order.
The order is equal to $p - 1$ exactly when $a$ is a [[primitive root|0KM2CJ80G7]] modulo $p$.
For example, modulo $13$, the order of $3$ is $3$ and the order of $2$ is $12$. Both divide $12$.

## Explanations

### L1 human

The powers of $a$ modulo $p$ go round a cycle. The length of the cycle always divides $p - 1$.
Modulo $11$, the powers of $5$ are $5, 3, 4, 9, 1$. The cycle has length $5$, and $5 \mid 10$.