The Euclidean Algorithm is a very powerful device. We will apply it to three problems:

We will show how the proofs are created and how they are polished for the final version. Hopefully, it will show that there is a huge difference between the way a proof is created and how it is presented.

The Division Lemma

The obvious way to find the gcd of two integers is to factorize them – we know we can do this by the Fundamental Theorem of Arithmetic (Theorem 25.5) – and find what common prime factors they have. Multiplying these together gives the gcd. For example, 440 = 23 × 5 × 11 and 1300 = 22 × 52 × 13. The common factors are 22 and 5 so the gcd is 22 × 5 = 20.

The problem is that factorization is hard for large numbers. Instead of this brute force method we shall give a more useful method. We start with a lemma.