Application of LCM and HCF

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Definition of LCM and HCF

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 What are applications of lcm and hcf, and the relation between hcf and lcm which will make the concept more clear.


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LCM and HCF Relation

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  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


\[\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{GCD/HCF of denominators}}\]


\[\text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}\]


What are Applications of LCM and HCF

We Use H.C.F. Method in Following Fields

  1. To split things into smaller sections.

  2. To equally distribute any number of sets of items into their largest grouping.

  3. To figure out how many people we can invite.

  4. To arrange something into rows or groups.

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 Following Fields :

  1. About an event that is or will be repeating over and over.

  2. To purchase or get multiple items in order to have enough.

  3. To analyse when something will happen again at the same time.

Real Life Example: Ram exercises every 8 days and Deepika every 4 days. Ram and Deepika both exercised today. After how many days 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

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 and 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

  1. 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: 6 students).

  2. Find the Highest Common Factor of 18, 24 and 42.(answer: 6).


Fun Facts

Euclid developed the method for finding HCF.

FAQ (Frequently Asked Questions)

1. What is the Prime Number?

Answer: A number which is divisible by only 1 and itself are called prime numbers. It has only two factors that is 1 and the number itself.


Consider this number: 12. This number can be found in many multiplication tables for example

1 x 12 = 12.

2 x 6 = 12

3 x 4 = 12


That means, 12 has many factors (1,2,3,4,6,12). Such a number is called a composite number.


On the other hand, consider this number: 29. As 29 is not divisible by any number, you will not find it in any table except 29 x 1 =29. Such a number is called a prime number.

2. How to Find LCM of Co-Prime Numbers.

Answer:

  • Co prime numbers are those numbers that don't have any common factors. For example, 14 and 15.

  • Individually none of them is prime because 14=2 x 7 and 15 = 3 x 5.

  • But they (14 and 15) do not have any common factors. But if they are given together they’re called co-prime numbers.

  • Any two consecutive numbers are co-prime numbers. (e.g. 11,12 or 1548,1549).

  • To find LCM of co-prime numbers, just multiply them and you will get LCM. There is no need to find factors. Example

  • Advantages of this method is it is extremely fast when you’ve to find LCMs of two digit numbers for example 12,15,96.

  • But it becomes tedious to find the LCM, as the number grows bigger, for example LCM (235, 512).