HCF

HCF is the abbreviation of the mathematical term Highest Common Factor. Historically mathematicians have set alternative names for this concept such as the Greatest Common Divisor (GCD) or the Greatest Common Measure (GCM). Although the term HCF is commonly used in mathematical problems, one must not get confused if they are asked to evaluate the GCD or GCM of two or more integers as all of them mean the same thing. The greatest factor that can divide the other values is termed HCF. It can either completely divide the other numbers producing a quotient of 1 or partially divide them producing values that are further divisible. The HCF value can never be zero, as it is not possible to divide any real number with 0.

Let us understand with one example. The Highest Common Factor of 60 and 15 is 5 as 5 is the largest integer that can divide both the numbers. Though 60 and 15 may also be divided by 3, we’ll not consider it because 5 > 3.

HCF Calculator

HCF can be evaluated for a specific group of numbers by applying two popular methods. They are as follows:

• Division method
• Prime factorization method

How to find the HCF by division method

This technique deals with the division of the numbers that are listed in the problem. By following this pattern, we find the possible common factors for each number. In the last step, we compare the factors to see which is the greatest.

Example: Find the HCF of 24 and 36.

Let us discuss each step of the division method.

1. Note down the numbers 24 and 36 beside each other putting a comma in between.
2. Divide both the numbers with the smallest possible prime integer, here it is 2. The result must come as a whole number with no remainder.
3. In the next line list the quotients similarly separating them with commas. After we divide 24 and 36 with 2, we will get 12 and 13 respectively. 
4. Repeat the steps from 1 to 3 till the point where there is no common factor.
5. On the left side of the numbers, we will get the factors one after the other. The product of the factors will be the HCF. For example, here the HCF is 2 itself as the numbers 12 and 13 cannot be further divided by a common number. 

How to find the HCF by prime factorization method 

In the previous section, we discussed the evaluation of HCF by division. Now let us implement the prime factorization technique to reach the desired outcome. Let us understand the procedure with an example. 

Example: Find the HCF of 120, 144, and 216.

Here are the necessary steps:

1. We have to multiply the prime factors of each number to represent them as a product value. 

Let us do this. 

120 = 23 x 3 x 5

140 = 22 x 5 x 7

216 = 23 x 33

1. In this step, we have to note down the common factors that divide each number. Case 2 is the only common factor. We will consider the lowest power of 2 
2. All common factors are multiplied to determine the HCF. 

Here, the HCF is 22 = 4.

HCF Calculator – an alternative method

1. Consider dividing the larger integer with the smaller one. 
2. Now we can divide the smaller number or divisor of the first step by the resultant remainder. 
3. Similarly, keep on dividing the divisor of the previous step by the remainder. This is to be continued until there is no remainder left. 
4. The required HCF is the divisor of the final step. 

HCF calculation method followed for more than two integers 

We will understand with an example. 

Find the HCF of 20, 60, and 90.

In the first step, we will determine the greatest common divisor of 20 and 60. 

22 x 5 = 20

22 x 3 x 5 = 60

Thus, by prime factorization, we have determined that 22 x 5 = 20 is HCF of 20 and 60. It is not mandatory to use only the prime factorization in this step, one can also do the division method. 

Now we have to evaluate the HCF of 90 and the greatest common divisor of the first two integers. 

2 x 32 x 5 = 90

22 x 5 = 20

Thus 2 and 5 are the common factors. 

Therefore, the HCF of 90 and 20 is 10.

This HCF will become the greatest common measure of all the three integers. 

∴ HCF of 20, 60, and 90 is 10.

Conclusion 

HCF gives us factors that divide the given values exactly. A number that has no other factor than 1 is called a prime integer. HCF of a set of integers can never be a greater number when compared to them.