Highest Common Factor (H.C.F)

The Highest Common Factor has been used in the mathematical calculations that indicate the greatest factor occurs between multiple numbers. It has been used to split numerical sets of data into smaller numerals. HCF works by equally distributing different numbers into the possible largest sets which will provide the greatest numbers in a numerical set of data. Several ways have been formulated to determine the Highest Common Factor. However, three methods have been used to properly determine the factors of the numbers. This article will evaluate the context with its determination method.   

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Defining the Highest Common Factor

Every divisible number has factors that construct that number. The common numerical factors that build a relationship between the numbers are classified as common factors. The Highest Common Factors of two or multiple numbers are the numbers that are found to be present in every numerical factor of the numbers. The highest number thus is determined as the HCF. 

Example:

The Highest Common Factors of 40 and 24 are determined below:

40= 2×20= 4×10= 8×5= 2x2x2x5

24= 2×12= 4×6= 8×3= 2x2x2x3

In this mathematical operation, the common factors are 2,4, and 8. However, the biggest number among the factors is 8. Thus, the GCF between 40 and 24 is 8.

Properties of HCF

HCF has been considering one of the important aspects of mathematics. However, the properties of HCF are described below to determine its approaches.

• The Highest Common Factor of two or multiple numerical data can divide each other without forming any remainder. 
• The HFC of multiple numerical sets of data is the common factor that can be obtained by each numerical set of data. 
• The HCF of two or multiple numbers will be equal to or less than each numeric data. 
• The Highest Common Factor of two or multiple numerical prime data will always be 1. 

Calculation of HCF

Several methods have been used to determine the Highest Common Factor of numerical data. However, every possible method represents the same HCF as a result. The most common methods that have been used to calculate the HCF of two numerical data are:

Listing Factor Method: This is one of the easiest methods to calculate the Highest Common Factor of two numbers. This method follows the creation of a list of common factors that are present in the numbers. After that, the highest numbers among the list of common factors have been identified to calculate the mathematical problem. 

For example: Determine the HCF of two numbers 42 and 30.

Solution: The list of the factors has been formed. The list of factors of the number 30 are 1, 2,3,5,6,10,15,30. The factors of the number 42 are 1, 2,3,6,7,14,21,42. The list of the factors of these two numerical data is observed as 1,2,3,6. However, number 6 is determined as the biggest number among them. Thus, the HCF of this mathematical problem is 6. 

Prime Factorization Method: This method has been done by prime factorization in two steps. The first step is determining the common prime factors and the second step is to multiply those common prime factors to obtain the result.

For example: Determine the HCF of 90 and 60.

Solution: Prime factors of 90 are 2,3,3,5 and of 60 are 2,2,3,5. The common factors among the numbers are 2, 3, and 5. Thus the multiplication of the numbers that is 2x3x5=30 is the Highest Common Factor of the numbers 90 and 60.

Division Method: This method has also been used to determine HCF between two numbers. However, three steps have been observed to include in this method. The first step is dividing the larger number by the smaller number to get the remainder. The remainder thus can be used to divide the smallest number in the second step. The last step will be followed by continuing the sequence until the remainder is obtained as 0. The last divisor will be the HCF of those two numbers. 

For example, The HCF of 360 and 198 will provide the number 18 as the last divisor. Thus the HCF will be 18. 

Conclusion

The Highest Common Factor has been used in various mathematical problems to simplify the equations. The usefulness of the HCF has been observed to play a major role in difficult equations at all levels of mathematical complications. The simplifications made by this method can be done by various methods on two or multiple numbers. Moreover, Highest Common Factor has a relationship with other methods of mathematical equations like the Least Common Multiple. In addition, this method has contributed to the development of mathematics.