In mathematics, (GCD) the greatest common divisor of two or more integers, that are not all zeros, is the largest positive integer which divides each of the integers.
The greatest common divisor can also be referred to as the: greatest common factor (gcf), greatest common measure (gcm), highest common factor (hcf) or highest common divisor.
Let’s begin with an example,
What is the greatest common factor of 12 and 16?
SOLUTION
That is, 12: 1, 2, 3, 4, 6 and 12
For 16, we have, 1, 2, 4, 8 and 16
FACTOR
Factors are numbers that can be multiplied together in order to get another number: 2 × 3 =, both 2 and 3 are factors. A number can have many factors: for example, the factors of 12 are 1, 2, 3, 4, 6 and 12. This is due to the fact that 1 x 12 = 12, 2 x 6 = 12 and 3 x 4 = 12.
COMMON FACTOR
Assuming that the factors of two numbers have been worked out, for example, the factors of 12 and 30:
Factors of 12 are 1, 2, 3, 4, 6 and 12
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
Common factors are those that appear in both the lists.
The following is an example having three numbers. What are the common factors of 15, 30 and 105?
Factors of 15 are 1, 3, 5 and 15
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105
The factors that appear in all the three lists are 1, 3, 5 and 15. Therefore, the common factors of 15, 30 and 105 are 1, 3, 5 and 15.
GREATEST COMMON FACTOR
This simply refers to the largest of the common factors. For example, the greatest common factor in the previous example of 15, 30 and 105 is 15.
USES
The main use of the greatest common factor is in simplifying fractions. For example, in case you are asked to simplify the fraction \(^{12}/_{30}\), start by finding the greatest common factor. The greatest common factor is 6 and therefore, we can divide both 12 and 30 by 6. 12 ÷ 6 = 2 and 30 ÷ 6 = 5. Therefore, the fraction \(^{12}/_{30}\) can be simplified to \(^2/_5\).
The greatest common factor can also be found by finding the prime factors and combining the common ones together. For example, if you are supposed to find the greatest common factor of 24 and 108,
24 = 2 x 2 x 2 x 3
108 = 2 x 2 x 3 x 3 x 3
The common ones are 2 x 2 x 3. Therefore, the greatest common factor is 12.
