Euclid's lemma is a generalisation of Proposition 30 of Book VII of Euclid's Elements. The lemma states that Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Egypt during the early 3rd century BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. ...

If a positive integer divides the product of two other positive integers, and the first and second integers are coprime, then the first integer divides the third integer.

This can be written in notation: Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... Coprime - Wikipedia /**/ @import /skins-1. ...

If n|ab and gcd(n,a)=1 then n|b.

Proposition 30, also known as Euclid's first theorem, states: The three letter acronym GCD may refer to: Greatest common divisor — in mathematics Great circle distance — in navigation Griffith College Dublin — private college in Ireland Grand Comic-Book Database — database of comic book information Global Communications Devices — supplier of semiconductor devices used in wireless networking Gardner Carton & Douglas — a US... Euclid Euclid of Alexandria (Greek: ) (ca. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...

If a prime number divides the product of two positive integers, then the prime number divides at least one of the positive integers.

That can be written as: A prime number (or a prime) is a natural number that only has trivial divisors. ...

If p|ab then p|a or p|b.

Often times, proposition 30 is called Euclid's lemma instead of the generalisation. A lemma is a "mini" theorem that is proven and used to prove a bigger theorem. Most of the time in mathematics textbooks Euclid's lemma is used to prove the fundamental theorem of arithmetic. In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. ... In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. ...

Proof of Proposition 30

Say p is a prime factor of ab, but also state that it is not a factor of a. Therefore, rp = ab, where r is the other corresponding factor to produce ab. As p is prime, and also because it is not a factor of a, a and p must be coprime. This means that two integers x and y can be found so that 1 = px + ay (Bézout's identity). Multiply with b on both sides: Coprime - Wikipedia /**/ @import /skins-1. ... In number theory, Bézouts identity, named after Étienne Bézout, is a linear diophantine equation. ...

b = b(px + ay)

b = bpx + bay.

We stated previously that rp = ab, and so:

b = bpx + rpy

b = p(bx + ry).

Therefore, p is a factor of b. This means that p must always exactly divide either a or b or both. Q.E.D. Q.E.D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, which was to be demonstrated). This is a translation of the Greek (hóper édei deĩxai) which was used by many early mathematicians including Euclid and Archimedes. ...

See also

• Euclidean algorithm
• Fundamental theorem of arithmetic