Euclid's lemma
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Euclid's lemma is a generalisation of Proposition 30 of Book VII of Euclid's Elements. The lemma states that
- 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:
- If n|ab and gcd(n,a)=1 then n|b.
Proposition 30, also known as Euclid's first theorem, states:
- If a prime number divides the product of two positive integers, then the prime number divides at least one of the positive integers.
Which can be written as:
- 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.
Proof of Proposition 30
Say p is a prime factor of ab, but also state that it is not a factor of a. Therefore, <math>rp = ab<math>, 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 <math>1 = px + ay<math> (Bézout's identity). Multiply b on both sides:
<math>b = b(px + ay)<math>
<math>b = bpx + bay<math>
We stated previously that <math>rp = ab<math>, and so:
<math>b = bpx + rpy<math>
<math>b = p(bx + ry)<math>
Therefore, p is a factor of b. This means that p must always exactly divide either a or b or both.
QED.