The highest common factor is the factor that is greatest between two or more numbers; some numbers may have more than one common factor, and in this state, the largest number would be the greatest common factor.
The following is an example of factors of 12 and 16:
Factors of 12: 1,2,3,4,6,12
Factors of 16: 1,2,4,8,16
Now some of the factors in the above factorization are the same, but 4 is the highest common factor between these two numbers.
The highest common factor is one of the most important areas of mathematics, which starts in the fourth grade and extends to higher education.
Finding the greatest common factors of two numbers might be easy, but for more than two numbers it could be a lengthy process, which can also be a complex question. There are several ways to calculate the highest common factor of several numbers. In this article, we would discuss some of the possibilities of calculation including the manual way and through the calculator.
Why do we need to learn the biggest common factor?
Teachers and students usually learn and teach the methods to find HCF, but most of them do not know why they are learning. There are various applications of the highest common factor that are helpful in professional life.
Some of the uses of common factors, including the highest and least common factors, are explained below:
The highest common factor is usually used to create the fraction in the lowest terms, while the LCM is helpful when making an addition as opposed to fractions.
For example: 18 / 48 = 18 ÷ 6 / 48 ÷ 6 = 3 / 8 (GCF)
The highest and lowest common factors are also helpful in algebra. In this area, the GCF is of crucial importance for the students because it enables them to know the GCF of the coefficients.
For example: 18×2 + 48x = 6x 3x + 8 (GCF)
The biggest common factor is also helpful in reducing the problems for students in education, especially in mathematics. Not only this, but this term is also helpful in solving the problems of problem-solving algorithms. It is also observed that this term helps students to increase their understanding.
Some examples of how this can help students are given below:
- Samantha has pieces of fabric. One piece is 72 inches in size, and the alternative piece is 90 inches in size. She wishes to reduce each portion into strips of equal width, which could be as extensive as possible. How extensively do the stripes have to be?
- Bennie prepares a party and puts snacks on plates. He has seventy-two cheese leaves and forty-eight carrot sticks. He needs each meal on each plate. He wants to distribute the meals calmly and he doesn’t need any leftover. What is the largest amount of plates he can use? and how many of each form of food does he have to place on each plate?
Methods of finding the highest common factor
One of the easiest ways to find the highest common factor of multiple numbers is the factorization method. First of all, you should list out all the factors of each number for which you are finding. You can also use the factors calculator for finding the factors instantly within moments.
For the whole number factors, the remainder would be zero because of division evenly. Once you find the common factors of all the numbers, compare it, and find the greatest number between the list. This way, you would find the greatest common factor of different numbers.
Here is the example of finding through factorization:
The factors of 20 are 1, 2, 4, 5, 10, 20
The factors of 28 are 1, 2, 4, 7, 14, 28
Now the common factors between two numbers are 1, 2, and 4. The greatest common factor between 20 and 28 is 4.
2. Prime Factorization
Another method of finding the highest common factor is Prime Factorization which is also an easy task. To start with it, you need to find the prime factors of each number whether there are two or more numbers. The numbers can be repeated while only write the prime factors which are common to the original number.
Now multiply the number that is common as well as has higher occurrence and this is the highest common factor. Here is the example of finding the HCF of 18 and 27 through prime factorization:
The prime factors of 18 are 2 x 3 x 3 = 18
The prime factors of 27 are 3 x 3 x 3 = 27
The highest common occurrences of these numbers are 3, 3.
So, the highest common factor is 3 x 3 = 9.
3. Euclid’s Algorithm
This is a popular method of finding the highest common factor of multiple numbers especially for the numbers that are very large like 199388, 2323523, and 2352225. Although it’s possible within seconds through the calculators doing it manually might make some complexity in it. Following is the way of finding the HCF of large numbers manually:
- From the two numbers, subtract the small number from the large number and note down the result.
- Repeat the process by subtracting the small number from the number until the result becomes smaller than the original small number.
- In every step, the result would be the new large number.
- Repeat the process until you reach the remainder as zero.
- Once you reach the zero, go to the step before zero and this is the greatest common factor.
Here is the example of finding the HCF of 18 and 27 through Euclid’s algorithm:
27 – 9 = 9
18 – 9 – 9 = 0
9 will be the highest common factor because it was the smallest number until you reach 0.
There is a total of six methods to find the highest common factor but three of the above-mentioned ways are the easiest and most common ones.