## Introduction

For the better understanding of the concept LCM (Lowest Common Multiple) and HCF (Highest Common Factor), we need to recollect the terms multiples and factors. Let’s learn about LCM, HCF, and relation between HCF and LCM of natural numbers.

**Multiples:** A multiple is any number which is exactly divisible by a given number. Ex: 3, 6,9,12, etc are the multiples of 3.

**Factors**: A factor is a number which divides any given number without leaving a remainder. Ex: 2,3,4,6,8,12 are the factors of 24.

**Lowest Common Multiple (LCM): **The least or smallest common multiple of any two or more given natural numbers are termed as **LCM**. For example, LCM of 10, 15, and 20 is 60.

**Highest Common Factor (HCF): **The largest or greatest factor common to any two or more given natural numbers is termed as **HCF** of given numbers. Also known as GCD (Greatest Common Divisor).For example, HCF of 4, 6 and 8 is 2.

4 = 2 × 2

6 =3 × 2

8 = 4 × 2

Here, highest common factor of 4, 6 and 8 is 2.

Both HCF and LCM of given numbers can be found by using two methods; they are division method and prime factorization.** **

**Properties of HCF and LCM**

**Property 1: **The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers.

LCM × HCF = Product of the Numbers

Suppose A and B are two numbers, then.

LCM (A & B) × HCF (A & B) = A × B

**Example 1:** **Prove that: LCM (9 & 12) × HCF (9 & 12) = Product of 9 and 12**

**Solution: **LCM and HCF of 9 and 12:** **

9 = 3 × 3 = 3²

12 = 2 × 2 × 3 = 2² × 3

LCM of 9 and 12 = 2² × 3² = 4 × 9 = 36

HCF of 9 and 12 = 3

LCM (9 & 12) × HCF (9 & 12) = 36 × 3 = 108

Product of 9 and 12 = 9 × 12 = 108

Hence, LCM (9 & 12) × HCF (9 & 12) = 108 = 9 × 12

**Property 2: **HCF of co-prime numbers is 1. Therefore LCM of given co-prime numbers is equal to the product of the numbers.

LCM of Co-prime Numbers = Product Of The Numbers

**Example 2: 8 and 9 are two co-prime numbers. Using this numbers verify, LCM of Co-prime Numbers = Product Of The Numbers**

**Solution: **LCM and HCF of 8 and 9:

8 = 2 × 2 × 2 = 2³

9 = 3 × 3 = 3²

LCM of 8 and 9 = 2³ × 3² = 8 × 9 = 72

HCF of 8 and 9 = 1

Product of 8 and 9 = 8 × 9 = 72

Hence, LCM of co-prime numbers = Product of the numbers

**Property 3: **H.C.F. and L.C.M. of Fractions

LCM of fractions = \(\frac{LCM \: of\: numerators}{HCF \: of\: denominators}\)

HCF of fractions = \(\frac{HCF \: of\: numerators}{LCM \: of\: denominators}\)

**Example 3:** **Find the HCF of \(\frac{12}{25}\), \(\frac{9}{10}\), \(\frac{18}{35}\), \(\frac{21}{40}\)**

**Solution:** The required HCF is = \(\frac{ HCF\: of\: 12, \: 9,\: 18, \: 21}{LCM \: of\: 25, \: 10,\: 35, \: 40}\) = \(\frac{3}{1400}\)

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