The greatest feasible number that divides both numbers perfectly is the highest common factor (HCF) of two numbers. HCF denotes the HCF of a and b. (a, b). If d is the HCF (a, b), you won’t be able to locate a common factor of a and b bigger than d. The greatest common divisor (GCD) is another name for the highest common factor (HCF) (GCD)
There are several methods for calculating the HCF of two numbers. One of the quickest approaches to determine the HCF of two numbers is to factorise each number into its prime components, and then multiply the product of the least powers of the common prime factors to get the HCF of those numbers.
Examine the numerous characteristics and properties of HCF to gain a better understanding of it. Find answers to problems like what is the highest common factor for a group of numbers, how to calculate HCF quickly, the relationship between HCF and LCM, and more. Continue reading! Let’s take a closer look at HCF.
Definition of HCF
The HCF (Greatest Common Factor) of two or more numbers is the highest number among all common factors of the provided numbers, or, to put it another way, the HCF (Highest Common Factor) of two natural numbers x and y is the largest integer that divides both x and y. Let’s look at this definition in terms of two numbers: 18 and 27. 1, 3, and 9 are the common factors of 18 and 27. The number 9 is the highest (biggest) of them. As a result, the HCF of 18 and 27 equals 9. HCF (18,27) = 9 is how it’s written.
Examples of HCF
Using the definition above, the HCF of:
- 60 and 40 is 20, i.e, HCF (60,40) = 20
- 100 and 150 is 50, i.e, HCF (150,50) = 50
- 144 and 24 is 24, i.e, HCF (144,24) = 24
- 17 and 89 is 1, i.e, HCF (17,89) = 1
How do I locate HCF?
The highest common factor of the provided numbers can be found in a variety of methods. The solution to the HCF of the numbers would always be the same, regardless of the method. The HCF of two numbers can be calculated using one of three methods:
- HCF by putting together a list of the most common factors
- prime factorization HCF
- HCF using the division method
Let’s take a closer look at each method by looking at the examples.
HCF by Making a List of Common Factors
We enumerate the factors of each number and determine the common factors of those numbers using this procedure. The highest common factor is then determined among the common factors. Let’s look at an example to better grasp this procedure.
The HCF of 30 and 42 will be discovered. We’ll go over the 30 and 42 factors. 1, 2, 3, 5, 6, 10, 15, and 30 are the factors of 30; 1, 2, 3, 6, 7, 14, 21, and 42 are the factors of 42. The common factors of 30 and 42 are clearly 1, 2, 3, and 6. However, the most common and largest component is 6. As a result, the HCF of 30 and 42 equals 6.
Prime Factorization HCF
We find the prime factorization of each integer and the common prime factors of those numbers using this method. The HCF of those numbers is then calculated by multiplying the prime factors that are common to all of the specified numbers. Let’s look at an example to better grasp this procedure.
The HCF of 60 and 90 will be discovered. Let’s use prime factorization to represent the numbers. So, 60 = 2 2 3 5 and 90 = 2 3 3 5 are the answers. HCF 60 and 90 will now be the product of common prime factors 2, 3, and 5. As a result, the HCF of 60 and 90 = 2 3 5 = 30.
HCF Calculated by Division
We divide the larger number by the smaller number and check the remaining in this approach. The remainder of the previous step is then used as the divisor, and the divisor of the previous step is used as the dividend, and the long division is repeated. We continue long division until the remainder equals 0, and the final divisor is the HCF of those two values. Let’s look at this method with the help of several examples.
Let’s use the division method to calculate the HCF of 198 and 360. The greater of the two figures presented is 360, while the smaller number is 198. We examine the residual after dividing 360 by 198. The remainder is 162 in this case. Repeat the long division with the remainder 162 as the divisor and the divisor 198 as the dividend. This method will be repeated until the remainder is 0 and the last divisor is 18, which is the HCF of 198 and 360.
Multiple Numbers HCF
Finding the HCF of many numbers follows the same steps as finding the HCF of a single number using the methods “HCF by listing out the common factors” and “HCF by prime factorization.” The method for calculating the HCF of several numbers using the division method, on the other hand, is different.
HCF of three different numbers
Follow the instructions below to find the HCF of three digits.
- We’ll start by calculating the HCF of two of the numbers.
- The HCF of the third number and the HCF of the first two numbers will be determined next.
Find the HCF of 126, 162, and 180.
First, we will find the HCF of the two numbers 126 and 180
Thus, HCF of 126 and 162 = 18. Next, we will find the HCF of the third number, which is 180, and the above HCF 18.
HCF of 126, 162 and 180 = 18