Lowest Common Multiple (LCM):
The smallest number (other than zero) that is the common multiple of any two or more given natural numbers are termed as LCM. For example, LCM of 6, 8, and 12 is 24.
Highest Common Factor (HCF):
The greatest factor which is 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 8,and 40 is 8.
Both HCF and LCM of given numbers can be found by using two methods, they are division methods and prime factorization.
HCF and LCM have many applications in our daily life. Let us understand the applications of LCM and HCF, also we will understand the relationship between these two, which will make the concept more clear.
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Relationship Between LCM and HCF
LCM and HCF have an interesting correlation between them. Some of LCM and HCF relations are as follows:
Relation 1: The product of LCM and HCF of any two given numbers is equivalent to the product of the given numbers.
LCM × HCF = Product of the Numbers
Suppose P and Q are two numbers, then.
LCM (P & Q) × HCF (P & Q) = P × Q
Relation 2: As HCF of co-prime numbers is 1. Therefore we get that the LCM of given co-prime numbers is equal to the product of the numbers.
LCM of Co-prime Numbers = Product of The co-prime Numbers.
Relation 3: H.C.F. and L.C.M. of Fractions
= LCM of fractions= LCM of numerators/ (GCD/HCF of denominators)
= HCF of numerators =HCF of fractions / LCM of denominators
What are the Applications of LCM and HCF?
We use H.C.F. method in the following fields:
Real Life Example: Priyanka has two pieces of cloth. One piece is 45 inches wide and the other piece is 90 inches wide. She wants to cut both the strips of equal width. How wide should she cut the strips?
Answer:
This problem can be solved using H.C.F. because we are cutting or “dividing” the strips of cloth into smaller pieces (Factor) of 45 and 90 (Common) and we are looking for the widest possible strips (Highest)
So,
H.C.F. of 45 and 90
45 = 3 x 3 x 5
90 = 2 x 3 x 3 x 5
HCF is 3 x 3 x 5 = 45
So we can say that
Priyanka should cut each piece to be 45 inches wide.
We use L.C.M. method in the following fields :
Real Life Example: Ram exercises every 8 days and Deepika every 4 days. Ram and Deepika both exercised today. After how many days do they exercise together again?
This problem can be solved using Least Common Multiple because we are trying to find out the time they will exercise, time that it will occur at the same time (Common).
Answer:
L.C.M. of 8 and 4 is
8 = 2 x 2 x 2
4 = 2 x 2
LCM is 2 x 2 x 2 = 8
SO, they will exercise together again in 8 days.
Solved Examples on LCM and HCF
1. Find the HCF of the following numbers:
36
48
60
Solution:
36 = 2 x 2 x 3 x 3
48 = 2 x 2 x 2 x 2 x 3
60 = 2 x 2 x 3 x 5
HCF(36, 48, 60)= 2 x 2 x 3 = 12
Therefore HCF of 36, 48, 60 is 12
2. Find the LCM of the following numbers:
25
40
Solution:
25 = 5 x 5
40 = 2 x 2 x 2 x 5
LCM = 5 x 2 x 2 x 2 x 5 = 200
Therefore LCM of 25 and 40 is 200
Quiz Time
Mr Patil has three classes. Each class has 28, 42 and 56 students respectively. Mr Patil wants to divide each class into groups so that every group in every class has the same number of students and there are no students left over. What is the maximum number of students Mr. Patil can put into each group?
(answer: 14 students).
Find the Highest Common Factor of 18, 24 and 42.
(answer: 6).
Fun Facts
Tricks for finding the HCF and LCM easily :
Trick 1 –
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Observe the numbers and list them out.
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Express each of them as a product of prime factors using prime factorization. For example, if we take the number 18 , it can be broken as 2 × 3². Similarly, factorise each of the numbers.
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Now, the product of all these highest powers of prime factors will be equal to the LCM.
Trick 2 –
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Note the numbers in a horizontal line and separate them with the help of commas.
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Take the smallest prime number and divide the given number from it.
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Note the undivided numbers and quotients in a line as well.
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Repeat this process until there is no prime factor common between two numbers.
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Then, multiply all the divisors and the numbers left in the last row. The final product of these numbers will be equal to the LCM of those numbers.
Trick 3 – How to Find LCM of Co-Prime Numbers
Hint: This question is using the concept of LCM as well as co-prime numbers. Co-prime numbers are the numbers having 1 as a common factor. Use this property to get find their LCM and hence the solution to this question:
Complete step-by-step answer:
As we know, the two numbers which have only 1 as their common factor are known as co-primes.
For example, Factors of 5 are 1 and 5
Factors of 3 are 1 and 3.
Here, the common factor is 1.
Thus, 5 and 3 are the co-primes.
Now, to find out the LCM of two or more numbers using the factorization method, we have to find their factors. Then common will be taken as LCM.
We have to choose each prime number with the greatest power and then we have to multiply them to get their LCM. So, in the above example case,
LCM of 3 and 5 will be,
3 × 5 = 15
Hence, we c
an say that the LCM of two co-prime numbers is nothing but their product.
∴ L.C.M. of two or more co-prime numbers is their product.
Note: Here two terms must be clear for getting a solution. One is LCM and the other is Co-primes. Least Common Multiple i.e. LCM is a method to find the minimum common multiple between any two or more numbers. LCM denotes the least value of common factor or multiple of any two integers. Co-prime number is a set of numbers that have only 1 as their common factor, which means their HCF will be 1.Then after this question it is very easy to get a solution.
Thank you, This was all about Applications of LCM and HCF. provides courses which include lectures, PDF notes, question and MCQ series etc to provide better and easy learning for students. Students can study more math topics further with the help of this course.