- Euclid's lemma
Euclid's lemma (Greek "polytonic|λῆμμα") is a generalization of Proposition 30 of Book VII of "
Euclid's Elements". The lemma states that:If a positive integerdivides 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::If "n"|"ab" and
gcd("n","a") = 1 then "n"|"b".
Proposition 30, also known as
Euclid's first theorem, states::If a prime numberdivides the product of two positive integers, then the prime number divides at least one of the positive integers.That can be written as::If "p"|"ab" then "p"|"a" or "p"|"b".Often, proposition 30 is called Euclid's lemma instead of the generalization. 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.
Proof of Proposition 30
Say "p" is a prime factor of "ab", but also state that it is not a factor of "a". Therefore, , 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 ( Bézout's identity). Multiply with "b" on both sides:
We stated previously that , and so:
Therefore, "p" is a factor of "b". This means that "p" must always exactly divide either "a" or "b" or both.
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).
Fundamental theorem of arithmetic
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