Smallest prime factor

Definition

Let $n$ be a natural number with $n \ge 2$. The smallest prime factor of $n$ is the smallest prime number $p$ such that $p \mid n$. It exists by the existence of a prime divisor.

Remarks

The smallest prime factor of $n$ is also the smallest divisor $d$ of $n$ with $d > 1$. The number $n$ is prime if and only if its smallest prime factor is $n$. If $n$ is composite, its smallest prime factor $p$ satisfies $p^2 \le n$. See a composite number has a prime factor at most its square root. For example, the smallest prime factor of $91 = 7 \cdot 13$ is $7$.

Explanations

L1 human

Divide $n$ by $2$, then by $3$, then by $4$, and so on. The first number that divides $n$ exactly is the smallest prime factor. It is always prime, because a smaller factor of it would also divide $n$ and would come earlier.