Lemat Euklides

Euclid's Lecture - Generalization of Theorem 30 of the Book of Elements Euclid. The content of the lemma is as follows: If a natural number divides the product of two certain natural numbers and is relatively prime to one, then it is a divisor of the other.

This can be saved as: n | a b & # x2227; dd ( n , a ) = 1 & # xA0; ⇒ & # xA0; n | b {\displaystyle n|ab\land {\mbox{nwd}}(n,a)=1\ \Rightarrow \ n|b} , where nwd is the largest common divisor.

Euclid's lemma is often used with to the theorem, not its generalization. Theorem states that if the first number divides the product of two natural numbers, then divides at least one of them: p | a b ⇒ p | a & # x2228; p | b {\displaystyle p|ab\Rightarrow p|a\lor p|b}

The above property characterizes the first numbers and is the motivation of the definition of the first ideal. Theorem 30 and its generalization are mainly used in the theory of numbers, especially in the proofs of the fundamental theorem of arithmetic.

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