Prime factorization

Definition

Let $n$ be a positive integer. A prime factorization of $n$ is a finite list of prime numbers $p_1 \le p_2 \le \dots \le p_k$ such that $n = p_1 p_2 \cdots p_k$. For $n = 1$, the prime factorization is the empty list, with the empty product $1$.

Remarks

We often group equal primes and write $n = q_1^{e_1} q_2^{e_2} \cdots q_r^{e_r}$, with distinct primes $q_1 < \dots < q_r$ and exponents $e_i \ge 1$. For example, the prime factorization of $360$ is $2, 2, 2, 3, 3, 5$, that is, $360 = 2^3 \cdot 3^2 \cdot 5$. Each positive integer has a prime factorization, and the fundamental theorem of arithmetic says that it is unique.

Explanations

L1 human

A prime factorization breaks a number into primes, which you cannot break further. A factor tree does this: $360 = 36 \cdot 10 = (6 \cdot 6) \cdot (2 \cdot 5) = 2 \cdot 3 \cdot 2 \cdot 3 \cdot 2 \cdot 5$. Sort the primes to get the list $2, 2, 2, 3, 3, 5$.