Introduction

We will look at two important properties of positive numbers namely:

1. The Euclidâ€™s division algorithm
2. Fundamental Theorem of Arithmetic

Euclid's division algorithm, deals with divisibility of integers.

The Euclidean algorithm is the oldest algorithms known, since it appeared in around 300 BC in Euclid's 7th book â€˜Elementsâ€™ , Proposition 2.

Euclidâ€™s Elements

The theorems says that any positive number

a can be divided by another positive number b in such a way to leave a remainder r smaller than b. This result can be recognized with the usual long division method. This theorem gains its relevance, due to its applications related to divisibility properties of integers.

According to the fundamental theorem of Arithmetic every composite number can be expressed as a product of primes in a unique way. This theorem is used to explore when the decimal expansion of a rational number say is terminating and when it is non-terminating repeating.