A Brief Introduction to HCF
The Highest Common Factor (HCF) of two numbers is the highest possible number that divides both numbers completely. The Highest Common Factor (HCF) is also called the Greatest Common Divisor (GCD). The HCF of two numbers can be determined by a variety of methods. Using the prime factorisation method is one of the easiest ways to get the HCF of two or more numbers. In this article, we will learn about HCF and how to find it. Also, we will see the HCF chart.
A number completely divides the given number without leaving any remainder is said to be the factor of that number. A number can have either positive or negative factors. For example, let's find the factors of 8. Since the number 8 is completely divisible by 1, 2, 4, and 8, we can list these as the positive factors of 8. The product of two negative numbers is a positive number, hence (-1) (-8) = 8 and (-2) (-4) = 8. In addition, the number 8 has negative factors, which can be written as -1, -2, -4, and -8.
When two or more numbers are exactly divided by the same number(s), those common divisors are known as common factors of the given numbers. Or we can say a common factor is a number that divides a pair of two or more numbers exactly without leaving any remainder. For example, the factors of 18 are 1, 2, 3, 6, 9, and 18.
Let's find out the common factors of the given numbers: 8, 12, 15, and 18.
Step 1: First of all we will find out the factors of the given numbers.
Factors of 8 = 1, 2, 4, and 8.
Factors of 12 = 1, 2, 3, 4, 6, and 12.
Factors of 15 = 1, 3, 5, and 15.
Factors of 18 = 1, 2, 3, 6, 9, and 18.
Step 2: Now, we will check the common factors (8, 12, 15, 18) = 1
So, 1 is the common factor of 8,12,15, and 18.
What is HCF?
The HCF (Highest Common Factor) of two or more numbers is the highest number among all the common factors of the given numbers. In simple words, the HCF (Highest Common Factor) of two natural numbers x and y is the largest possible number that divides both x and y.
Using the above definition, the HCF of a few sets of numbers can be listed as follows:
HCF of 60 and 40 is 20, i.e., HCF (60, 40) = 20
HCF of 100 and 150 is 50, i.e., HCF (150, 50) = 50
HCF of 144 and 24 is 24, i.e., HCF (144, 24) = 24
HCF of 17 and 89 is 1, i.e., HCF (17, 89) = 1
Steps to Find HCF
Let us understand the steps to find the HCF using two numbers by using the prime factorisation method:
What is the HCF of 8 and 12?
Step 1: First of all, find factors of 8 and 12 separately.
Factors of 8 = 1, 2, 4, and 8.
Factors of 12 = 1, 2, 3, 4, 6, and 12
Step 2: Now, find the common factors of 8 and 12.
The common factors of 8 and 12 are 1, 2, and 4.
Step 3: Now check the highest number among the common factors.
Among these numbers, 4 is the highest (largest) number.
So, the HCF of 8 and 12 is 4.
This is written as HCF (8, 12) = 4.
Observe the following HCF chart to understand this concept.
HCF of 8 and 12
Q1. Find the HCF of 24 and 32 by prime factorisation.
Ans: Prime factorisation of 24 and 32 is $(2 \times 2 \times 2 \times 3)$ and $(2 \times 2 \times 2 \times 2 \times 2)$ respectively.
As visible, 24 and 32 have common prime factors.
Hence, the HCF of 24 and 32 is $2 \times 2 \times 2=8$.
Q2. What is the HCF of 20 and 30?
Ans: Factors of 20: 1, 2, 4, 5, 10, 20
Factors of $30: 1,2,3,5,6,10,15,30$
There are 4 common factors of 20 and 30, which are $1,2,10$, and 5.
Therefore, the highest common factor of 20 and 30 is 10.
Q1. What is HCF of 135 and 225?
Q2. What is HCF of 65 and 117?
From the HCF of two numbers examples, we have learned to find the factors and then take the common factors. Then after that multiplication of the common factors will give us the HCF. We can use the division method to find the HCF. Hence, after going through this article we have made our concepts of HCF very clear in both methods and now we know how to find the HCF of two numbers. With the examples and practice problems, we will get more clarity on the topic. So after reading the article, go through the practice problem to have a better understanding.
FAQs on HCF of 8 and 12
1. What are the methods to find the HCF?
The following are the methods to find the HCF:
Prime Factorisation Method
Long Division Method
Listing Common Factors
2. What is the HCF of two consecutive natural numbers?
Two consecutive numbers always have an HCF of one. Since there is just one common element between the two successive numbers, this is the case. So 1, between two consecutive numbers, becomes the HCF.
For instance; Take the numbers 5 and 6 in consecutive order. The only factor that connects the two numbers is 1, and nothing else.
Thus, the HCF is 1. This demonstrates that any pair of consecutive numbers' HCF is always one.
3. What is the difference between HCF and LCM also find the LCM and HCF of 12, 15 and 21?
The H.C.F. defines the greatest factor present in between given two or more numbers, whereas L.C.M. defines the least number which is exactly divisible by two or more numbers. H.C.F. is also called the greatest common factor (GCF) and LCM is also called the Least Common Divisor.
HCF of 12, 15, and 21 is 3, whereas the LCM of 12, 15, and 21 is 420.