# Euclid's lemma

In algebra and number theory, **Euclid's lemma** is a lemma that captures a fundamental property of prime numbers, namely:^{[note 1]}

**Euclid's lemma** — If a prime *p* divides the product *ab* of two integers *a* and *b*, then *p* must divide at least one of those integers *a* and *b*.

For example, if *p* = 19, *a* = 133, *b* = 143, then *ab* = 133 × 143 = 19019, and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact, 133 = 19 × 7.

If the premise of the lemma does not hold, i.e., *p* is a composite number, its consequent may be either true or false. For example, in the case of *p* = 10, *a* = 4, *b* = 15, composite number 10 divides *ab* = 4 × 15 = 60, but 10 divides neither 4 nor 15.

This property is the key in the proof of the fundamental theorem of arithmetic.^{[note 2]} It is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings. Euclid's Lemma shows that in the integers irreducible elements are also prime elements. The proof uses induction so it does not apply to all integral domains.

The lemma first appears as proposition 30 in Book VII of Euclid's *Elements*. It is included in practically every book that covers elementary number theory.^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}

The generalization of the lemma to integers appeared in Jean Prestet's textbook *Nouveaux Elémens de Mathématiques* in 1681.^{[9]}

In Carl Friedrich Gauss's treatise *Disquisitiones Arithmeticae*, the statement of the lemma is Euclid's Proposition 14 (Section 2), which he uses to prove the uniqueness of the decomposition product of prime factors of an integer (Theorem 16), admitting the existence as "obvious". From this existence and uniqueness he then deduces the generalization of prime numbers to integers.^{[10]} For this reason, the generalization of Euclid's lemma is sometimes referred to as Gauss's lemma, but some believe this usage is incorrect^{[11]} due to confusion with Gauss's lemma on quadratic residues.

In modern mathematics, a common proof involves a result called Bézout's identity, which was unknown at Euclid's time.^{[12]} Bézout's identity states that if *x* and *y* are relatively prime integers (i.e. they share no common divisors other than 1 and -1) there exist integers *r* and *s* such that

Let *a* and *n* be relatively prime, and assume that *n*|*ab*. By Bézout's identity, there are *r* and *s* making

The first term on the left is divisible by *n*, and the second term is divisible by *ab*, which by hypothesis is divisible by *n*. Therefore their sum, *b*, is also divisible by *n*. This is the generalization of Euclid's lemma mentioned above.

The following proof is inspired by Euclid's version of Euclidean algorithm, which proceeds by using only subtractions.

One has to prove that n divides b. For proving this by strong induction, we suppose that this has been proved for all positive lower values of ab.

The positive integers *a* – *n* and n are coprime: if any prime divides both, then it divides their sum, and thus divides both n and a. This is a contradiction of the coprimality hypothesis. As a consequence of the right hand side
, *q* – *b* is positive. So, the conclusion follows from the induction hypothesis, since *a* – *n* < *a*.

Euclid's lemma is proved at the Proposition 30 in Book VII of Euclid's *Elements*. The original proof is difficult to understand as is, so we quote the commentary from Euclid (1956, pp. 319–332).