Question

State Euclid's Lemma and state its uses.

Answer 1

In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers.

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.

The lemma is not true when p is a composite number. 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.

Answer 2

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