ICSE Class 8 Maths Number Systems Chapter 06 HCF _ LCM

HCF And LCM

Factorisation of Numbers

Finding the HCF

Finding the LCM

LCM and HCF of Two Numbers

LCM and HCF of Coprime Numbers

Introduction

A number that divides another number exactly, without leaving any remainder, is known as a factor of the second number.

\(n_1 \times n_2 = n_3\)

As factor \(n_1\) multiplied by factor \(n_2\) results in \(n_3\) as the product, the number \(n_3\) is known as a multiple of its factors.

The first counting number or 1 has only one factor in itself.

All other natural numbers have two or more factors.

Prime numbers are natural numbers which have only 1 and the number itself as their factors.

Set P = \(\{2, 3, 5, 7, 11, \ldots\}\)

Even numbers are natural numbers that are multiples of 2.

Set E = \(\{2, 4, 6, 8, 10, \ldots\}\)

Odd numbers are natural numbers that do not have 2 as a factor.

Set O = \(\{1, 3, 5, 7, 9, 11, \ldots\}\)

Composite numbers are natural numbers which have at least one factor, other than 1 and the number itself.

Set C = \(\{4, 6, 8, 9, 10, 12, \ldots\}\)

Two natural numbers are known as coprime numbers when they do not have any factors in common, other than 1.

\(C_P = \{\{5, 7\}, \{9, 10\}, \{14, 15\}, \ldots\}\)

Factorisation of Numbers

Set A = \(\{x \mid x = \frac{72}{a}, a \in \mathbb{N}, x \in \mathbb{N}\}\)

Representing set A in Roster form, we have:

72 is not exactly divisible by any other value of a.

Thus, set A = \(\{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\}\).

To find all the factors of a number, divide it by the smallest natural number, till the divisor is less than the quotient. All the divisors and quotients will be the factors of the number.

Prime Factorisation

The expression of a number as a product of its prime factors is called its prime factorisation.

Example 1: 210 = \(2 \times 3 \times 5 \times 7\)

Example 2: 10296 = \(2 \times 2 \times 2 \times 3 \times 3 \times 11 \times 13\) = \(2^3 \times 3^2 \times 11 \times 13\)

Teacher's Note

Understanding prime factorisation helps us decompose numbers into their simplest building blocks, similar to how we might break down complex tasks into simpler steps in our daily work.

Common Factors

Set A = \(\{x \mid x = \frac{168}{a}, a \in \mathbb{N}, x \in \mathbb{N}\}\)

= \(\{1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168\}\)

Set B = \(\{x \mid x = \frac{144}{b}, b \in \mathbb{N}, x \in \mathbb{N}\}\)

= \(\{1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144\}\)

\(A \cap B = \{1, 2, 3, 4, 6, 8, 12, 24\}\), which are the common factors.

The highest common factor (HCF) of 168 and 144 is 24.

Common Multiples

Set A = \(\{x \mid x = 6a, a \in \mathbb{N}, x \in \mathbb{N}\}\)

= \(\{6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, \ldots\}\)

Set B = \(\{x \mid x = 8b, b \in \mathbb{N}, x \in \mathbb{N}\}\)

= \(\{8, 16, 24, 32, 40, 48, 56, 64, 72, 80, \ldots\}\)

\(A \cap B = \{24, 48, 72, \ldots\}\), which are the common multiples.

The lowest common multiple (LCM) of 6 and 8 is 24.

Teacher's Note

Common factors and multiples appear in everyday situations like scheduling events or dividing items equally among groups.

Finding The HCF

Prime Factorisation Method

Write down the common prime factors of the given numbers. The HCF will be the product of these common prime factors.

Example 3: Find the HCF of 216, 360, and 432.

Prime factorisation of 216 = \(2 \times 2 \times 2 \times 3 \times 3 \times 3\)

Prime factorisation of 360 = \(2 \times 2 \times 2 \times 3 \times 3 \times 5\)

Prime factorisation of 432 = \(2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3\)

The product of the underlined common factors = \(2 \times 2 \times 2 \times 3 \times 3\) = 72

Thus, the HCF of 216, 360, and 432 is 72.

Cross Division Method

The Cross Division Method is used to find the HCF for large numbers, when finding all their prime factors is a lengthy process.

HCF Of Two Numbers

Divide the greater number by the smaller number and find the remainder. Take the remainder as the new divisor and the old divisor as the new dividend and divide. Continue this process till the remainder is 0. The last divisor is the HCF of the two numbers.

Example 4: Find the HCF of 1144 and 1287.

Step 1:

1144 \(\mid\) 1287

-1144

143

Step 2:

143 \(\mid\) 1144

-1144

0

143, being the last divisor, is the HCF of 1144 and 1287.

Teacher's Note

The Euclidean algorithm for finding HCF is like the process of finding common ground in discussions - we keep narrowing down until we reach the core agreement.

HCF Of Three Numbers

Find the HCF of any two numbers. Then find the HCF of the third number and the HCF already obtained.

Example 5: Find the HCF of 6300, 7560, and 8820.

Step 1:

6300 \(\mid\) 7560

-6300

1260

Step 2:

1260 \(\mid\) 6300

-6300

0

Thus, the HCF of 6300 and 7560 is 1260.

Step 3:

1260 \(\mid\) 8820

-8820

0

Thus, the HCF of 6300, 7560, and 8820 is 1260.