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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.
There are two methods to find out the HCF of the given numbers, namely;
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
The division method is a shortcut method to find the HCF of two numbers.
Problem 2
Find out the HCF of 120 and 144 by the division method.
Solution 2
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.
There are two methods to find out the LCM of the given numbers, namely;
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
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
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