The Highest Common Factor or HCF of two numbers is defined as the highest common number that is present in both the numbers, which means each number may have many numbers that are factors of it. Two numbers may have many common factors between them, and HCF or Highest Common Factor is the highest value among them. It is also called either Greatest Common Factor (GFC) or Greatest Common Divisor (GCD). Zero can never be the highest common factor. When it comes to smaller numbers, for example, 8 and 4, we can easily understand that four would be the HCF. But, it is difficult to find through simple calculus when it comes to higher numbers, and we often use some methods to find them manually.
Two of the techniques that we would describe here are the prime factorization method and the long division method.
Prime Factorisation Method
To find the HCF of two numbers, first, we have to find the prime factors of each of the two numbers and write them as a product of these. After writing both numbers as a product of their prime factors, we have to identify the common prime factors among them. Multiplying them with the lowest degree can give HCF of the two numbers.
Illustrating the above method through a question can make this concept crystal clear.
Question: Find the HCF of 24 and 36.
Answer: Step 1:
Write the two numbers as a product of prime factors only. In that manner, we can write the numbers as,
24 = 2 x 2 x 2 x 3
36 = 2 x 2 x 3 x 3
From there, we understand that 2 x 2 x 3 is common.
Therefore, HCF of 24 and 36 is = 2 x 2 x 3 = 12.
Long Division Method
To find the HCF of two numbers using the long division method, first, we have to divide the large number with a small number. Once if a remainder is left after the division, we have to divide the first divisor with the remainder we got in the previous division, and if the remainder is precisely dividing the divisor completely, we have come to the conclusion that it is the HCF of the given two numbers. Otherwise, in case if the remainder does not divide the first divisor completely, we have to repeat the steps again to get the HCF of the two given numbers. Let us illustrate this method also through a question.
Question: What is the HCF of 120 and 100?
Answer: Let us solve this question through some steps.
First, let us divide the larger number 120 by, the smaller number 100.
120/100 gives one and a remainder of 20. As there is a remainder of 20, the first divisor, that is, 100, needs to be divided by the remainder of 20.
100/20 gives five and is left with no remainder. From this, we reach a conclusion that 20 is the HCF of 120 and 100.
Besides just a mathematical tool, HCF can find many applications in our daily life too. Dividing things into smaller sections, equal distribution of a set of the number of things are some examples where finding HCF can help out.
For example, 5 and 6 are said to be co-prime numbers as they are consecutive integers, and also their highest common factor is 1. Similarly, sets of numbers such as 3, 4, and 2, 3 are all different examples of co-prime numbers.
The highest Common factor or HCF is a simple yet very useful tool in mathematics. We should always make our basics clear so that mathematics can also be handled so easily. Mathematics can be an easy subject once the base is straightforward and robust. Therefore, we should become familiar with methods that help us find HCF and be thorough with it.
Q1. Besides in Daily Life Mathematics, How can Highest Common Factor or HCF Influence Concepts in Mathematics?
Ans: The highest common factor, more commonly called HCF, is a simple yet powerful tool in mathematics. Once mastered, the skill of finding HCF can help us in many ways. When we move on to higher classes, we may find complex questions whose first step involves finding HCF of any set of numbers. We could only continue with the question only if we know the proper method to find HCF. Moreover, HCF can be used to reduce fractions to simpler or smaller numbers so that the calculation part can be handled comparatively easily. HCF can also be used to find the value of another common mathematical tool, the Least Common Multiple or LCM.
Q2. How can we Find the HCF of Three Numbers? Is the Process the Same as that for Solving HCF Two Numbers?
Ans: The process for finding the HCF of three numbers is just the same as that for finding two numbers. Let us illustrate this by solving a question through the prime factorization method.
Exp: Find the HCF of 60, 96, and 126 through the prime factorization method.
Soln: Step 1: Write all the numbers as the product of prime numbers only
60 =2 x 2 x 3 x 5
96 =2 x 2 x 2 x 2 x 2 x 3
126 =2 x 3 x 3 x 7
The next step is to sort out prime numbers, which are common in all three numbers. According to that, we find that 2 x 3 is common in every number. Therefore HCF of 60, 96, and 126 is found to be 2 x 3 = 6.
Q3. How is HCF Applicable in Daily Life?
Ans: When there comes a condition where you have to divide something equally between some members, HCF serves the purpose well. HCF can be used mainly to divide things, whether it is to divide something correctly between people, share something properly, etc.