# Proof: By Euclid

(related to Proposition: 7.31: Existence of Prime Divisors)

• For since $A$ is composite, some number will measure it.
• Let it (so) measure ($A$), and let it be $B$.
• And if $B$ is prime then that which was prescribed has happened.
• And if ($B$ is) composite then some number will measure it.
• Let it (so) measure ($B$), and let it be $C$.
• And since $C$ measures $B$, and $B$ measures $A$, $C$ thus also measures $A$.
• And if $C$ is prime then that which was prescribed has happened.
• And if ($C$ is) composite then some number will measure it.
• So, in this manner of continued investigation, some prime number will be found which will measure (the number preceding it, which will also measure $A$).
• And if (such a number) cannot be found then an infinite (series of) numbers, each of which is less than the preceding, will measure the number $A$.
• The very thing is impossible for numbers.
• Thus, some prime number will (eventually) be found which will measure the (number) preceding it, which will also measure $A$.
• Thus, every composite number is measured by some prime number.
• (Which is) the very thing it was required to show.

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