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HCF and LCM - finding Methods, Prime factorisation method, Division method, formula and Applications

 

HCF

The full-form of HCF is the Highest Common Factor. It can be defined for two or more numbers, and it represents the greatest number that can divide these numbers.

Methods to find out HCF

There are two methods to find out the HCF of the given numbers, namely;

  1. Prime factorisation method
  2. Division method

Prime factorisation method

In this method, we take all the prime factors of all the given numbers. Then the common factors are multiplied to get HCF.

Problem 1

Find out the HCF of 120, 144, and 168 by prime factorisation.

Solution 1

First, let’s write the numbers in terms of the prime factors,
120= 2 x 2 x 2 x 3 x 5
144= 2 x 2 x 2 x 2 x 3 x 3
168= 2 x 2 x 2 x 3 x 7

The common factors are 2 x 2 x 2
Therefore, HCF = 2 x 2 x 2 =8

Division method

The division method is a shortcut method to find the HCF of two numbers.

  • In this method, the larger of the two numbers is divided by the smaller number.
  • In the second step, the smaller number is divided by the remainder of the first step.
  • Then the first step remainder is divided by the remainder we got now.

Problem 2

Find out the HCF of 120 and 144 by the division method.

Solution 2

image1

LCM

The full-form of LCM is Least Common Multiple. It can be defined for two or more numbers, and it represents the smallest number that can be divided by these numbers.

Methods to find out LCM.

There are two methods to find out the LCM of the given numbers, namely;

  1. Prime factorisation method
  2. Division method

Prime factorisation method

In this method, we take all the prime factors of all the given numbers. Then each factor is multiplied by itself based on the most number of times it occurs.

Problem 1

Find out the LCM of 45 and 60 by prime factorisation.

Solution 1

First, let’s write the numbers in terms of the prime factors,
45= 3 x 3 x 5
60= 2 x 2 x 3 x 5

2 occurs two times
3 occurs two times
5 occurs one time

Therefore, LCM = 2 x 2 x 3 x 3 x 5 = 180

Division method

In this method, we divide the numbers simultaneously by the prime factors. If a number is not divisible by the prime factor, leave it as it is.

Problem 2

Find out the HCF of 45 and 60 by the division method.

Solution 2

image2

HCF and LCM formula

Product of two numbers = (HCF of the two numbers) x (LCM of the two numbers)

Example:
Take the numbers 45 and 60
HCF = 15
LCM = 180

45 x 60 = 15 x 180 = 2700

Applications of HCF and LCM

  1. HCF and LCM are useful to split large numbers into small manageable fractions.
  2. It helps in the proper arrangement of number sets in rows and groups.
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