If you are here that means you know what least common multiple means but that is not all, there is another term known as the greatest common divisor.  The easiest definition of the greatest common divisor is the highest number which can divide two or more than two numbers. For example, we have two numbers that are 16 and 32, the objective is to find the greatest common divisor of both numbers. When we talk about divisors, there are many divisors of both numbers, let's write it down. The divisors of 16 are 2,4,8, and 16, and the divisors of 32 are 2,4,8,16, and 32. If we analyze both divisors then we have some common numbers and they are 2, 4, 8, and 16 now the question is which one is the greatest common divisor? Remember the condition for GCD? The number should be the highest and common of the selected numbers. We have gathered all the common divisors now its time to find the highest number. The number 16 is the highest number in that series therefore we can declare 16 as the GCD of 32, and 16.

That was easy, isn't it? But things get complicated when we try to find GCD of big numbers or many numbers. That is why we have a good method that can help you to calculate the greatest common divisor.

Calculation of the Greatest Common Divisor

Below are the steps to find the greatest common divisor.

Step 1

Decompose the number into prime numbers. For example, if you have a number 42, break it into its prime factors. The prime factors will be 2 \times 3 \times 7. This was pretty easy, sometimes it might take you towards long division.

Step 2

After the decomposition process, compile the prime numbers. Here is a small tip, if there are more than two same prime numbers then write it into its power form. This will help you in the next step.

Step 3 (Final Step)

Compare and find the common number to find the GCD. For example, you decomposed two numbers into its prime factors and here are the results, { 2 }^{ 3 } \times { 3 }^{ 1 } \times { 5 }^{ 2 } and { 2 }^{ 5 } \times { 5 }^{ 1 }. Now find common numbers with respect to its powers. In the above example, we can get { 2 }^{ 3 } and { 5 }^{ 1 } as common, take them out and expand them.

{ 2 }^{ 3 } = 8 \qquad { 5 }^{ 1 } = 5

8 \times 5

40

Hence, the GCR for the above example is 40.

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Examples

Determine the GCD of: 72, 108 and 60:

\begin{matrix} 2 \\ 2 \\ 2 \\ 3 \\ 3 \\  \\ \end{matrix}\begin{matrix} \left| 72 \right \\ \left| 36 \right \\ \left| 18 \right \\ \left| 09 \right \\ \left| 03 \right \\ \left| 01 \right \end{matrix} \qquad \begin{matrix} 2 \\ 2 \\ 3 \\ 3 \\ 3 \\  \\ \end{matrix}\begin{matrix} \left| 108 \right \\ \left| 054 \right \\ \left| 027 \right \\ \left| 009 \right \\ \left| 003 \right \\ \left| 001 \right \end{matrix} \qquad \begin{matrix} 2 \\ 2 \\ 3 \\ 5 \\ \\ \end{matrix}\begin{matrix} \left| 60 \right \\ \left| 30 \right \\ \left| 15 \right \\ \left| 05 \right \\ \left| 01 \right \end{matrix}

72 = {2}^{3} \times {3}^{2}

108 = {2}^{2} \times {3}^{3}

60 = {2}^{2} \times 3 \times 5

GCD (72, 108, 60) = {2}^{2} \times 3 = 12

12 is the largest number that divides 72, 108 and 60.

If a number is a divisor of another, then this number is the GCD.

The number 12 is a divisor of 36.

GCD (12, 36) = 12

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.