Proposition VII.31 — Any composite number is measured by some prime number.

Proof

Descend through divisors: a composite has a proper divisor, that divisor is either prime or composite; if composite, repeat. The descent must terminate (numbers are bounded below by 2), so a prime divisor exists.

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Depends on (2)

VII.11Definition VII.11A prime number is that which is measured by a unit alone.
VII.13Definition VII.13A composite number is that which is measured by some number.

Required by (dependents) (2)

VII.32Proposition VII.32Any number either is prime or is measured by some prime number.
IX.20Proposition IX.20Prime numbers are more than any assigned multitude of prime numbers.

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