Natural Number is Prime or has Prime Factor

Theorem

Let $a$ be a natural number greater than $1$.

Then either:

$a$ is a prime number

or:

there exists a prime number $p \ne a$ such that $p \mathop \backslash a$

where $\backslash$ means is a divisor of.

In the words of Euclid:

Any number either is prime or is measured by some prime number.

Proof

By definition of composite number $a$ is either prime or composite.

Let $a$ be prime.

Then the statement of the result is fulfilled.

Let $a$ be composite.

$\exists p: p \mathop \backslash a$

where $p$ is a prime number.

The result follows by Proof by Cases.

$\blacksquare$

Historical Note

This theorem is Proposition $32$ of Book $\text{VII}$ of Euclid's The Elements.