how can we prove Euclid's division lemma?
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In mathematics, Euclid's lemma is most important lemma as regards divisibility and prim numbers. In simplest form, lemma states that a prime number that divides a product of two integers have to divide one of the two integers. This key fact requires a amazingly sophisticated proof and is a wanted step in the ordinary proof of the fundamental theorem of arithmetic.
Euclid's lemma in plain language says: If a number N is a multiple of a prime number p, and N = a · b, then at least one of a and b must be a multiple of p. Say,
Obviously, in this case, 7 divides 14 (x = 2).