HCF and LCM are one of the common terms in mathematics. Maths is never an easy subject to understand, it takes patience and a keen sense of curiosity and appreciation to comprehend it. But at your age, it is easy enough to grasp these concepts. Hence it’s of great importance that one spends enough time on revising and clearing their doubts.
Factors and multiples must have surely been covered in the previous classes. There is no need to be apprehensive. One only needs to understand the concepts clearly, in time, one will be able to tackle problems related to it quite easily.
HCF and LCM Formula
HCF (Highest Common Factor):
As has been taught in Factors and multiples, the factors of a number are all the numbers that divide into it. Let’s proceed to the highest common factor (HCF) and the least common multiple (LCM). As the rules of mathematics dictate, the greatest common divisor or the gcd of two or more integers, when at least one of them is not zero, happens to be the largest positive that divides the numbers without a remainder. For instance, take 8 and 12, the gcd of these two numbers or the HCF of two numbers will be 4. Since this greatest common divisor or GCD is also known as the highest common factor or HCF.
LCM- Least Common Multiple:
In arithmetic, the least common multiple or LCM of two numbers, let’s assume a and b, is denoted as LCM (a,b) is the smallest or least, as the name suggests, a positive integer that is divisible by both a and b. Take the LCM of 4 and 6. Multiples of four are: 4,8,12,16,20,24 and so on while that of 6 is 6,12,18,24…. The common multiples for four and six are 12,24,36,48…and so on. The least common multiple in that lot would be 12. Let us now try to find out the LCM of 24 and 15.
LCM of 24 and 15 = 2 × 2 × 2 × 3 × 5 = 120
How to find HCF and LCM?
Division method to find the HCF (Shortcut method)
Steps to find the HCF of any given numbers;
|1) Larger number/ Smaller Number|
|2) The divisor of the above step / Remainder|
|3) The divisor of step 2 / Remainder. Keep doing this step till R = 0(Zero).|
|4) The last step’s divisor will be HCF.|
The above steps can also be used to find the HCF of more than 3 numbers.
Suppose there are two numbers, 8 and 12, whose LCM we need to find. Let us write the multiples of these two numbers.
8 = 16, 24, 32, 40, 48, 56, …
12 = 24, 36, 48, 60, 72, 84,…
You can see, the least common multiple or the smallest common multiple between the two number, 8 and 12 is 24.
LCM of Fractions
To calculate the LCM of two numbers 60 and 45. Out of other ways, One way to find the LCM of given numbers is as below:
- List the prime factors of each number first.
60 = 2 × 2 x 3 × 5
45 = 3 × 3 × 5
- Then multiply each factor the most number of times it occurs in any number.
If the same multiple occurs more than once in both the given numbers, then multiply the factor the most number of times it occurs.
The occurrence of Numbers in the above example:
2: two times
3: two times
5: one times
LCM = 2 × 2 x 3 × 3 × 5 = 180
In BYJU’S you can also learn, Prime Factorization Of Hcf And Lcm.
HCF and LCM Questions
Example: Find the Highest Common Factor of 25, 35 and 45.
Solution: Given, three numbers as 25, 35 and 45.
25 = 5 × 5
35 = 5 × 7
45 = 5 × 9
From the above expression, we can say 5 is the only common factor for all the three numbers.
Therefore, 5 is the HCF of 25, 35 and 45.
Example: Find the Least Common Multiple of 36 and 44.
Solution: Given, two numbers 36 and 44.
Let us find out the LCM, by division method.
Therefore, LCM(36, 44) = 2 × 2 × 3 × 3 × 11 = 396
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