explain Euclid's lemma in detail
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In number theory, Euclid's lemma is a lemmathat captures a fundamental property of prime numbers, namely:[note 1]
Euclid's lemma — If a prime pdivides 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.
Inherently, 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 usesinduction so it does not apply to all integral domains.
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