## Introduction

On a Shatabadi Train, there are two types of coaches - Chair Car (CC), and Executive Class (EC). Tickets in the EC are priced almost double of the tickets in the CC. Each EC coach accommodates 46 passengers, while a CC coach can seat 72 passengers. **What is the LEAST number of coaches of each kind required to accommodate an equal number of passengers in each class?**

To maintain physical distancing the railway authorities at a particular station announced that from each coach an equal number of passengers will disembark at a time. **What is the MAXIMUM number of people that can disembark from each coach?**

Problems such as these involve two very useful and related mathematical concepts. These are:

- Lowest common multiple (LCM), and
- Highest common factor (HCF) or Greatest common divisor (GCD).

In this chapter we will learn two methods of finding the LCM and HCF of two positive integers. The first method is based on **Euclid's Division Lemma** and the second method is based on the **Fundamental Theorem of Arithmetic**.

The Fundamental Theorem of Arithmetic is a very simple yet powerful result in mathematics. In this lesson, we will also learn how this theorem can help us to:

- Prove that square root of some positive integers are irrational numbers; and
- Determine the nature of decimal representation of rational numbers

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