Matches in DBpedia 2015-10 for { ?s ?p "In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: If a prime divides the product of two numbers, it must divide at least one of those numbers. It is also called Euclid's first theorem although that name more properly belongs to the side-angle-side condition for showing that triangles are congruent. For example, 133 × 143 = 19019, and since 19019 is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact, 133 = 19 × 7.This property is the key in the proof of the fundamental theorem of arithmetic. It is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings.The lemma is not true for composite numbers. For example, 4 does not divide 6 and 4 does not divide 10, yet 4 does divide their product, 60."@en }
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- Euclids_lemma abstract "In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: If a prime divides the product of two numbers, it must divide at least one of those numbers. It is also called Euclid's first theorem although that name more properly belongs to the side-angle-side condition for showing that triangles are congruent. For example, 133 × 143 = 19019, and since 19019 is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact, 133 = 19 × 7.This property is the key in the proof of the fundamental theorem of arithmetic. It is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings.The lemma is not true for composite numbers. For example, 4 does not divide 6 and 4 does not divide 10, yet 4 does divide their product, 60.".