Greatest Common Divisor (GCD) Study Notes, Questions and Answers

The factors of 12 are 1,2,3,4,6, and 12. On the other hand, the factors of 18 are 1,2,3,6,9, and 18. The factors of 12 and 18 are the numbers that evenly divide them without leaving a remainder.
The common factors between the factors of both 12 and 18 are: 1, 2, 3, and 6, as shown below:
Math: GCD example

Therefore, the Greatest Common Divisor (GCD) of both 12 and 18 is 6.
The GCD is also referred to as the Greatest Common Factor (GCF), or the Highest Common Factor. Therefore, in Math questions, you can be asked to find the GCD, the GCF, or the HCF. All these terms refer to the same thing discussed here, and the approach of solving them is them same for all.

Finding the GCD

Option 1: To find the GCD of two or more numbers, first identify the factors of each of the numbers, then get the greatest common factor among them.
Option 2: Alternatively, express each number as product of its prime factor, and get the product of the common factors.
In our example above, we have used the first option as:
12: 1,2,3,4,6, and 12.
18: 1,2,3,6,9, and 18.
The common factors are 1, 2, 3, and 6. Hence, the GCD is 6.

Alternatively, we can do as follows:
Math: GCD example

Then we get the product of the common factors; 2 and 3 (2 × 3 = 6). Hence the GCD by option 2 is 6.

Practice Question 1: Find the GCD

Find the GCD of 73, 96, 300

Math: GCD example 1

Then we get the product of the common factors; 22 and 3 (2 × 2 × 3 = 12). Hence the GCD by option 2 is 12. Try solving it using the first option. Hint: You should still get 12 as your GCD.

Practice Question 2: Find the Greatest Number

Find the greatest number which, when divided by 181 and 236, leaves a remainder of 5 in each case.

First, we subtract 5 from each number to give:
181 - 5 = 176
236 - 5 = 231
Lets find the GCD of 176 and 231
Math: GCD example 1

The common factor of both 176 and 231 is 11.

Practice Question 3: Application of GCD in Real Life

Three similar iron bars of length 200cm, 300 cm, and 360 cm are cut into equal pieces. Find the largest possible area of a square which can be made from any of the three pieces.

First, we get the GCD.
Math: GCD example 2

Then we get the product of the common factors; 22 and 5 (2 × 2 × 5 = 20). Therefore, the largest common length of the 3 bars is 20cm.
Area of a square is L × L; where L = 20;
Area = 20 × 20 = 400cm2

Notes and Questions Related to Greatest Common Divisor (GCD)

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