# 1. Numbers and algebra

- Peter M. Higgins

### Abstract

‘Numbers and algebra’ introduces the number system and explains several terms used in algebra, including natural numbers, positive and negative integers, rational numbers, number factorization, the Fundamental Theorem of Arithmetic, Euclid’s Lemma, the Division Algorithm, and the Euclidean Algorithm. It proves that any common factor *c* of *a* and *b* is also a factor of any number of the form *ax + by*, and since the greatest common divisor (gcd) of *a* and *b* has this form, which may be found by reversing the steps of the Euclidean Algorithm, it follows that any common factor *c* of *a* and *b* divides their gcd *d*.