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Maths Query > Unit > Arithmetic > Number System

# Highest Common Factor or GCF. Properties of HCF

Found in topics: Factorization

## Introduction

HCF (Highest Common Factor) or GCF (Greatest Common Factor) or GCD (Greatest Common Divisor), they all are the same terms. We use them while finding out the number which is the largest common divisor among many divisors of a number.

To see in more detail how it is calculated, let’s review first some general terms related to HCF.

## Factor of a number

A factor of a number is that number which divides the number exactly.

Example

What are the factors of 40?

There are a total of eight factors of 40 viz. 1, 2, 4, 5, 8, 10, 20 and 40.
Why?
∵ these all numbers give the remainder zero when 40 is divided by 1, 2, 4, 5, 8, 10, 20 and 40.
How?
$\frac{40}{1}=40$
$\frac{40}{2}=20$
$\frac{40}{4}=10$
$\frac{40}{5}=8$
$\frac{40}{8}=5$
$\frac{40}{10}=4$
$\frac{40}{20}=2$
$\frac{40}{40}=1$

## Common factors of numbers

The factors which are common to two or more numbers are called common factors.

Example

Let’s understand it by taking the two numbers as 4 and 10 and further, finding out their common factors.

Step1: find out the factors of 4 and 10 separately.
Factors of 4:
$\frac{4}{1}=4$
$\frac{4}{2}=2$
$\frac{4}{4}=1$
There are a total of three factors of 4 viz. 1, 2 and 4.
Factors of 10:
$\frac{10}{1}=10$
$\frac{10}{2}=5$
$\frac{10}{5}=2$
$\frac{10}{10}=1$
There are a total of four factors of 10 viz. 1, 2, 5 and 10.
Step2: Find out the common factors of 4 and 10.
Finally, we can say 1 and 2 are the common factors of 4 and 10, because the factors 1 and 2 do exist for both numbers 4 and 10.

## How to find HCF?

The number which is greatest among the common factors is called the highest common factor (HCF) of two or more numbers.

Example

Let’s learn HCF as an example.

Find the HCF of 12 and 20.
So, what are the steps?
Step 1: Find out the factors for 12.
Step 2: Find out the factors for 20.
Step 3: Find out the common factors of 12 and 20.
Step 4: Find out the greatest number among those common factors, that will be the HCF of 12 and 20.
Step1: Find out the factors for 12.
$\frac{12}{1}=12$
$\frac{12}{2}=6$
$\frac{12}{3}=4$
$\frac{12}{4}=3$
$\frac{12}{6}=2$
$\frac{12}{12}=1$
There are a total of six factors of 12 viz. 1, 2, 3, 4, 6 and 12.
Step2: Find out the factors for 20.
$\frac{20}{1}=20$
$\frac{20}{2}=10$
$\frac{20}{4}=5$
$\frac{20}{5}=4$
$\frac{20}{10}=2$
$\frac{20}{20}=1$
There are a total of six factors of 20 viz. 1, 2, 4, 5, 10 and 20.
Step3: Find out the common factors of 12 and 20.
∴ the common factors are 1, 2 and 4.
Step4: Find out the greatest number among those common factors, that would be the HCF of 12 and 20.
∴ the greatest number among the common factors 1, 2 and 4 is 4.
Hence, HCF or GCF or GCD of 12 and 20 is 4.

## Five properties of HCF

1. The HCF of two or more numbers exactly divides the numbers.
Example

HCF of two numbers 20 and 50 is 10.
It means the HCF 10 also divides the two numbers 20 and 50.

2. The HCF of given numbers can’t be greater than its numbers .
Example

HCF of two numbers 27 and 60 is 3.
It means the HCF 3 can’t be greater than the two numbers 27 and 60

3. If one number is factor of another number, the smaller number will be HCF.
Example

HCF of two numbers 27 and 54 is 27.
Here, 2 and 27 are factors of number 54. Also, 27 number is the number for which HCF will be calculated.
Therefore, HCF will be the smallest number of 27 and 54, which is 27.

4. HCF of coprimes numbers is 1.
Example

HCF of 14 and 17 is 1.
Because, 14 and 17 are coprimes numbers.

5. HCF of consecutive numbers is always 1.
Example

HCF of 19 and 20 is 1.
Because, 19 and 20 are consecutive numbers.

## What is perfect number?

If the sum of all the factors of a number is two times the number itself, then the number is called a perfect number.

Example

6 is a perfect number

Why?
∵ sum of factors of 6 is 12
and 6 × 2 = 12
How?
Step1: Find out the factors of 6.
$\frac{6}{1}=6$
$\frac{6}{2}=3$
$\frac{6}{3}=2$
$\frac{6}{6}=1$
Factors of 6 are 1, 2, 3 and 6
Step2: Find the sum of factors
1 + 2 + 3 + 6 = 12
∴ we can see, the sum of factors of 6 which is 12 is equal to twice the number itself.

### 1) What is HCF?

HCF stands for Highest Common Factor. It is the largest value of factor that can be obtained from the common factors of two or more given numbers.

### 2) What are the methods to find HCF?

There are three methods to find HCF. 1)Common Factor Method 2)PrimeFactor Method 3)Division Method

### 3) What is GCD?

The Highest Common Factor is also called as Greatest Common Divisor (GCD). It is also known by other name GCF (Greatest Common Factor).

## Solved Examples

### 1) Find HCF of 70 and 85.

Factors of 70 are 1, 2, 5, 14, 35, 70.

Factors of 85 are 1, 5, 17.

From the above, the common factors of 70 and 85 are 1, 5.

Therefore, HCF 70 and 85 is 5.

### 2) Find HCF of 14, 24 and 36.

Prime factors of 14 = 2 × 7

Prime factors of 24 = 2 × 2 × 2 × 3

Prime factors of 36 = 2 × 2 × 3 × 3

Common prime factor = 2

∴ HCF of 14, 24 and 36 = 2

### 3) Find HCF of 70 and 385.

Factors of 70 are 1, 2, 5, 14, 35, 70

Factors of 385 are 1, 5, 7, 11, 35, 55, 77, 385

Common factors are 1, 5, 35

∴ HCF of 70 and 385 = 35

### 4) Find GCF of 30 and 110.

Factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30

Factors of 110 are 1, 2, 3, 5, 10, 11, 22, 55, 110

Common factors are 1, 2, 5, 10

∴ GCF of 30 and 110 = 10

### 5) Find GCF of 30 and 130.

Factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30

Factors of 130 are 1, 2, 5, 10, 13, 26, 65, 130

Common factors are 1, 2, 5, 10

∴ GCF of 30 and 130 = 10

### 6) The HCF of two numbers is 5. If the numbers are in the ratio 2:3. Find the numbers.

Let the numbers are 2x and 3x.
where HCF = x = 5
∴ The numbers are
2x = 2 × 5 = 10
3x = 3 × 5 = 15

### 7) The HCF of three numbers is 11. If the numbers are in the ratio 1:2:3. Find the numbers.

Let the numbers are x, 2x and 3x.
where HCF = x = 11
∴ The numbers are
x = 1 × 11 = 11
2x = 2 × 11 = 22
3x = 3 × 11 = 33

## Worksheet 2

### Multiple choice questions

a) 1

b) 0

c) 2

d) 3

a) 2

b) 4

c) 8

d) 1

a) 2

b) 5

c) 7

d) 1

a) 1

b) 2

c) 3

d) 0

a) 12

b) 24

c) 36

d) 1

a) 1

b) 2

c) 3

d) 4

a) 1

b) 2

c) 3

d) 4

a) 1

b) 2

c) 3

d) 4

a) 2

b) 3

c) 1

d) 6

#### 10) The HCF of 4 and 8 is

a) 4

b) 8

c) 2

d) 32

Last updated on: 30-07-2024