Prove Euclid's lemma.
Understand the Problem
The question is asking for a proof of Euclid's lemma, which states that if a prime number divides the product of two integers, then it must divide at least one of those integers. This involves concepts from number theory and the properties of prime numbers.
Answer
If \( p \mid ab \), then \( p \mid a \) or \( p \mid b \).
Answer for screen readers
If a prime number ( p ) divides the product ( ab ), then it must divide at least one of the integers ( a ) or ( b ).
Steps to Solve
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Start with the assumption
Assume a prime number ( p ) divides the product of two integers ( a ) and ( b ). In mathematical notation, this means ( p \mid ab ). -
Use the definition of prime numbers
Since ( p ) is a prime, it only has two positive divisors: 1 and ( p ) itself. This property will be crucial in our proof. -
Consider the case when ( p ) does not divide ( a )
If ( p ) does not divide ( a ), then ( a ) and ( p ) are coprime, meaning their greatest common divisor (gcd) is 1. This can be stated as ( \gcd(a, p) = 1 ). -
Apply Bézout's identity
By Bézout's lemma, since ( a ) and ( p ) are coprime, there exist integers ( m ) and ( n ) such that
$$ am + pn = 1 $$ -
Multiply through by ( b )
Now, multiply the entire equation by ( b ):
$$ b(am + pn) = b $$
This simplifies to
$$ abm + pbn = b $$ -
Analyze the resulting equation
From the original assumption ( p \mid ab ) and this new equation, we see that ( pbn ) is divisible by ( p ). Since ( p \mid b ) must hold true (the left side is divisible by ( p )), we conclude that ( p ) divides ( b ) when it does not divide ( a ). -
Conclude the proof
Thus, if ( p \mid ab ) and ( p \nmid a ), then ( p ) must divide ( b ). Given these cases, we have shown that if a prime ( p ) divides the product ( ab ), it must divide at least one of ( a ) or ( b ), thereby proving Euclid's lemma.
If a prime number ( p ) divides the product ( ab ), then it must divide at least one of the integers ( a ) or ( b ).
More Information
Euclid's lemma is a fundamental theorem in number theory and plays an important role in the proof of the unique factorization theorem, which states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors.
Tips
- Misunderstanding the properties of prime numbers, especially thinking that a prime may divide a product without affecting the individual factors.
- Failing to consider the case where the prime does not divide one of the integers, which is crucial for completing the proof.
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