TITLE: What is the importance of Bézout's identity?
QUESTION [14 upvotes]: I am creating a function which generates a privateKey for RSA. The underlying algorithm for generating a privateKey is the Extended Euclidean algorithm. According to Wikipedia, the output of this algorithm is a "Bézout's identity". 
I never heard of Bézout's identity before and wanted to know what it's importance is and what is it used for, but I can't find a clear answer. Googling "What is the importance of Bézout's identity?" yields no relevant results. The closest thing I could find was a discussion on Wikipedia:Talk

The point is that Bézout's identity is an important result which is
  used in many areas of mathematics. In particular it is one of the
  starting tools (with modular arithmetic) of Diophantine equation
  theory

To someone who does not have an extensive mathematical background the above discussion is meaningless. Can someone describe the importance and use-cases for Bézout's identity in layman terms?

REPLY [2 votes]: Another important use of Bézout's identity is in the proof of Euclid's Lemma:

If a prime $p$ divides the product $ab$ of two integers $a$ and $b$,
  $p$ must divide at least one of those integers $a$ and $b$.

Euclid's Lemma can be used to prove the fundamental theorem of arithmetic:

Every integer greater than 1 either is prime itself or is the product
  of prime numbers, and that this product is unique, up to the order of
  the factors.

As its name implies, the fundamental theorem of arithmetic is a cornerstone of number theory.