In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 is either prime or can be
In Disquisitiones Arithmeticae (1801) Gauss proved the unique factorization theorem and used it to prove the law of quadratic reciprocity.
Euclid's lemma
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In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers:
Fundamental theorem of arithmetic
…Proposition 30, (In modern terminology: if a prime p divides the product ab, then p divides either a or b or both.) Proposition 30 is referred to as Euclid's lemma, and it is the key in the proof of the fundamental theorem of arithmetic. Any composite number is measured by some prime number…
Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570.