This document discusses real numbers and Euclid's lemma. It provides definitions of real numbers, integers, rational and irrational numbers. It discusses mathematicians like Gauss who contributed to real numbers. Euclid's lemma states that if a prime number divides the product of two numbers, it must divide at least one number. The document compares division algorithms and provides examples of using Euclid's lemma to find the highest common factor and lowest common multiple of numbers. It also discusses prime and composite numbers.