How to prove Euclid's lemma?
Question:
How to prove Euclid's lemma?
Bézout's Identity:
Suppose that {eq}m {/eq} and {eq}n {/eq} are integers, and {eq}d=\gcd(m,n) {/eq}. Then Bézout's identity states that there are integers {eq}x {/eq} and {eq}y {/eq} with
{eq}mx+ny=k {/eq}
if and only if {eq}d|k {/eq}.
In particular, Bézout's identity means that the integers {eq}m {/eq} and {eq}n {/eq} are relatively prime (that is, {eq}\gcd(m,n)=1 {/eq}) if and only if there are integers {eq}x {/eq} and {eq}y {/eq} such that
{eq}ax+by=1 \, . {/eq}
Answer and Explanation: 1
Become a Study.com member to unlock this answer! Create your account
View this answerEuclid's lemma states that, if {eq}p {/eq} is prime and {eq}p|ab {/eq}, then either {eq}p|a {/eq} or {eq}p|b {/eq}.
To prove Euclid's lemma,...
See full answer below.
Learn more about this topic:
from
Chapter 1 / Lesson 11A linear combination of two integers can be shown to be equal to the greatest common divisor of these two integers. This is the essence of the Bazout identity. In this lesson, we prove the identity and use examples to show how to express the linear combination.