common factors

Highest common factors – Easy and Simple methods

The greatest common factor between two numbers is considered to be the greatest common factor. The highest common factor can be used in different areas, whereas it is normally taught from fourth grade up to the university studies.

The highest common factor can be used in mathematics, as students use it to clarify their complex questions, which are long and complex.

In this article, we will discuss some of the best methods to find the highest common factor.

Methods of finding the highest common factor

Although there are two methods of finding the highest common factor including the manual way and through an online calculator. The manual way further includes 6 ways but in online calculators automatically calculates. However, some of the online tools also calculate the result according to your desired method. 

The methods of finding the highest common factor are the following: 

  • Factorization for finding a highest common factor

To find the greatest common factor, one of the easiest methods is factorization. When factorizing, you should first list all the factors of each number, whether it is two or more numbers. Now, there may be some common factors between the numbers for which you find the number, and compare both lists of factors.

Now that the highest number between the factors has been compared, the highest number that is common is considered the highest common factor. Here is an example of how to find the highest common factor of 8 and 12, and 48 and 72.

HCF of 8 and 12 by factorization:

The factors of 8 are 1, 2, 4, 8

The factors of 12 are 1, 2, 3, 4, 6, 12

Now the common factors between these two numbers are: 1, 2, 4

But the highest common factor is 4. 

HCF of 48 and 72 by factorization:

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The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 15, 24, 48

The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

The common factors between these two numbers are: 1, 2, 3, 4, 6, 8, 12, 24

24 being the highest, is considered as the greatest common factor.

  • Prime Factorization 

This is another method to find the HCF between multiple numbers. In this way, you have to make a list of the prime factors of all the numbers for which you are finding. List down the prime factors that are common with every of the original number. Now, among these prime factors, take out the common occurrences that are also highest. Multiply these numbers and the result would the highest common factor. Here is the example of finding the GCF through prime factorization. 

HCF of 18 and 27 by prime factorization:

The prime factorization of 18 are 2 x 3 x 3 = 18

The prime factorization of 27 are 3 x 3 x 3 = 27 

The common occurrences between the factors of 18 and 27 are 3 and 3. 

The greatest common factor is 3 x 3 = 9. 

  • Euclid’s Algorithm 

Sometimes, you come into the situation that you have to find the highest common factor of the numbers that are complex and high. For example, it would be difficult for you to find the HCF of 234234, and 345364. For this, Euclid’s algorithm method would be helpful to find the HCF easily. 

However, the following is the way to find through this method: 

  • Out of the two numbers, subtract smaller out of the larger number and note the result. 
  • Now subtract the smaller number out of the resulting number and repeat the result until the result becomes smaller than the original smaller number.
  • After the result is smaller than the original smaller number, use the original smaller number as the new larger number and subtract the result out of the new larger number. 
  • Repeat the process until the remainder becomes zero. 
  • After reaching zero, go back a step and the number is the highest common factor. 

Here is an example: 

HCF of 8 and 12 by Euclid’s algorithm:

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12 – 8 = 4

4 – 4 = 0

So, the HCF of 8 and 12 through Euclid’s algorithm is 4. 

Calculating the Highest Common Factors using online calculators

The above-mentioned methods are just about the manual way of calculating while it can be complex and cause human or calculation errors. However, numerous online calculators could be used to find the HCF within seconds over a single click. 

The online calculators are usually available for free and you just need to enter the value and click to get all of the factors including the highest one. Here are some of the top calculators: 

  • MeraCalculator.com

(https://www.meracalculator.com/math/hcf.php

This is one of the pioneer calculators that are available online for free. The GCF calculator by the Meracalculator.com is efficient and popular because it displays all of the six methods at the same time after you enter the numbers. However, one of the other major advantages of using this calculator is that it gives the result in the actual syntax. This way, students can use this to write the factorization in the standard way and according to the syntax. 

Here is the picture of how it tells the factors with the syntax: 

  • Byjus.com 

Another popular website for providing mathematics especially for the HCF calculation is Byjus.com which is also free to use. The calculator is very fast and gives the result instantly but it doesn’t tell the method or syntax which is used for making the result. 

Here is the example of how it shows the result simply: