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Number Theory
Sometimes called "Euclid's lemma" in textbooks when appearing before a proof of the
fundamental theorem of arithmetic. It states that if
p is a
prime number and
p|ab, then either
p|a or
p|b ("|" means "divides").
Corollaries:
- If p is a prime and p|an, then p|a.
- If a and c are relatively prime, then c|ab implies c|b.
Incidentally, Euclid's Second Theorem states that
there are infinitely many primes.
References:
"Euclid's First Theorem" is sometimes referred as such according to MathWorld.com.
Many of these theorems appear in
Euclid's Elements.
Book VII,
proposition 30 states Euclid's First Theorem.
Book IX,
proposition 14 partially states the
fundamental theorem of arithmetic, and
proposition 20 states Euclid's Second Theorem.
Thanks to
Swap for tipping me off about this.