# Definition:Prime Decomposition/Multiplicity

## Definition

Let $n > 1 \in \Z$.

Let:

- $n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$

be the prime decomposition of $n$, where:

- $p_1 < p_2 < \cdots < p_r$ are distinct primes
- $k_1, k_2, \ldots, k_r$ are (strictly) positive integers.

For each $p_j \in \left\{ {p_1, p_2, \ldots, p_r}\right\}$, its power $k_j$ is known as the **multiplicity of $p_j$**.