Below on this page, you have everything you need to learn what the greatest common divisor is and how we can determine it (GCD) for any two or more numbers without using any calculator! Of course, it’s nice to know that it doesn’t necessarily mean that any two numbers have to have a greatest common divisor! Some numbers have no common divisor at all, just as there are numbers that are divisible only by themselves and the number 1 – prime numbers.
In the text below, you can see examples in which the greatest common divisor of two or more numbers is calculated! In addition to the examples that you can see directly on this page, you are welcome to view the video examples that our team has prepared for our readers. Note that the abbreviation GCD we use in the text is an abbreviation for the greatest common divisor.
Greatest common divisor
The greatest common divisor is a number (divisor) of two or more numbers is the greatest of the common divisors by which any of the numbers can be divided, without obtaining a certain remainder!
The greatest common divisor can be determined in several ways without using a calculator or any other aid. If the numbers whose GCD we are looking for are smaller, then the simplest way to determine it is by using arrays of divisors for each of the numbers.
How To Determine The GCD By Forming A Sequence Of Divisors
By forming an array of divisors, the first, second, and third examples below on this page are solved. See how it’s done:
Example 1: Find the greatest common divisor of the numbers 12 and 20!
GCD(12;20) = ?
Let’s list all the divisors that are divisible by 12:
12:1; 2; 3; 4; 6; 12
Let’s now form the same sequence from the numbers that are divisible by the number 20:
20:1; 2; 4; 5; 10; 20;
If we compare the two sequences, it is noticeable that the common divisors of the numbers 12 and 20 are the numbers 1, 2, and 4. Of these three numbers, 4 is the largest, so we conclude that the following applies:
GCD(12;20) = 4
Continue with the second example:
Example 2: Find the greatest common divisor of the numbers 18 and 36!
GCD(18;36) = ?
First we form a sequence of numbers that are divisible by the number 18:
18:1; 2; 3; 6; 9; 18
We form the same sequence for the number 36:
36: 1; 2; 4; 6; 9; 12; 18; 36
If we compare the two sequences, it is noticeable that the common divisors of the numbers 18 and 36 are the numbers 1, 2, 6, 9, and 18. Of these five numbers, 18 is the largest, so we conclude that:
GCD(18;36) = 18
How To Find Greatest Common Devisor For Three Numbers
As you can see, determining the greatest common divisor of two numbers is a very easy procedure. Let’s see how the GCD of three numbers can be determined:
Example 3: Find the greatest common divisor of the numbers 20, 25 and 30!
GCD(20;25;30) = ?
Of course, practicing the method we’ve already used in the first and second examples above, we proceed to form appropriate divisor sequences for all three numbers from the third example! These divisor sequences for the numbers 20, 25, and 30 are:
20:1; 2; 4; 5; 10; 20
25:1; 5; 25
30:1; 2; 3; 5; 6; 10; 15; 30
From the sequences shown above, it is easy to see that the common divisors of all three numbers are the numbers 1 and 5! Since 5 is greater than 1, we can conclude that:
GCD(20;25;30) = 5
For larger numbers, it is more difficult to form complete sequences with their divisors. For them, you can use the method of determining the greatest common divisor by factoring them! See the second method below:
How To Determine The GCD By Factoring
See example number 4:
Example 4: Find the GCD of the numbers 54 and 84!
GCD(54;84) = ?
If we want to avoid enumerating all the individual divisors of the numbers 54 and 84, we can represent them in the form of a product of their prime factors!
The number 54 represented as a product of its prime multiples looks like this:
54=2x3x3x3
So the number 84 represented as a product of its prime multiples looks like this:
84=2x2x3x7
From the contents of the common factors (from their prime factors), we can determine the greatest common divisor of the numbers 54 and 84. It is enough to calculate the product of each prime factor that appears as a factor in both the number 54 and the number 84. Finally, we draw the following conclusion from this rule:
GCD(54;84) = 2×3= 6
As always when you have more than one possibility in a particular mathematical calculation, it is up to you to decide which procedure you will use to determine the greatest common divisor of two or more numbers without any use of a calculator. Our recommendation to you is to use both methods depending on which exact numbers are in a specific example.
Finally, the video materials below contain many examples of determining the GCD of two or more numbers. We recommend you watch the video:
We wish you the best of luck in the future! All the best!
