Proposition IX.14 — If a number be the least that is measured by prime numbers, …

Proof

A least common multiple of primes equals their product; any prime dividing the product divides one of the factors (VII.30), hence is one of the originals. This is the kernel of unique factorisation.

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Depends on (2)

VII.30Proposition VII.30If two numbers by multiplying one another make some number, and any prime number measure the product, it will also…
VII.34Proposition VII.34Given two numbers, to find the least number which they measure.

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