# Introduction

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In the previous grades, we have encountered **Euclid’s division algorithm** and the **Fundamental Theorem of Arithmetic**.

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**Euclid’s division algorithm**, deals with divisibility of integers. Stated simply, it says:

**Any positive integer 'a' can be divided by another positive integer 'b' in such a way that it leaves a remainder 'r' that is smaller than b.**

Many of us recognise this as the usual **long division process**. Although this result is quite easy to state and understand, it has many applications related to the divisibility properties of integers. We touch upon a few of them and use it mainly to compute the

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The **Fundamental Theorem of Arithmetic**, on the other hand, has to do something with multiplication of positive integers. We already know that:

**Every number can be expressed as a product of primes in a unique way.**

This important fact is the Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and understand, it has some very deep and significant applications in the field of mathematics.

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We **use the Fundamental Theorem of Arithmetic for two main applications**:

**First, we use it to prove the irrationality of many of the numbers we have studied, such as:** ,2 and3 .5 **Second, we apply this theorem to explore when exactly the decimal expansion of a rational number, say** (where q ≠ 0) is terminating and when it is nonterminating- it is repeating.p q

We do so by looking at the prime factorisation of the denominator q of

So let's get started.