For integers $a$ and $b$, we say that $a$ divides $b$ when there is an integer $k$ such that

$$b=ak.$$

This is written

$$a\mid b.$$

Divisibility is a relation between integers, not ordinary division with a remainder.

A positive integer greater than $1$ is prime when its only positive divisors are $1$ and itself. A non-prime integer greater than $1$ is composite.

Every integer greater than $1$ can be written as a product of primes. Up to reordering, this prime factorization is unique.

Prime factorization exposes the multiplicative structure of integers and underlies greatest common divisors, modular arithmetic and much of elementary number theory.