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Greatest Common Factor or Gcf

Greatest Common Factor is the greatest number among the common factors of all the numbers. It is abbreviated as GCF.

When we list out the factors of two or more numbers, we might see many factors common to all the numbers. The greatest among them is called as greatest common factor.

For example, the factors of 24 are, 24, 12, 8, 6, 4, 2, 1 and the factors of 16 are 16, 8, 4, 2 ,1. The numbers in bold are the common factors of both 24 and 16 but 8 is the greatest among them. Therefore, 8 is the greatest common factor of 24 and 16.

This is one of the methods of finding the greatest common factor for a set of numbers.

Greatest common factor is also referred as greatest common divisor (GCD) or highest common factor (HCF).

An interesting fact is, the product of GCF and LCM of two (or more) numbers equals to the product of the numbers.

For example, the GCF and LCM of 24 and 16 are 8 and 48 respectively.

=> (8)(48) = 384, which is same as (24)(16) = 384

Prime Factorization

Another method of finding the greatest common factor of a set of numbers is by prime factorization.

For example, let us see what is the greatest common factor of 64 and 80.

64 = 2 x 2 x 2 x 2 x 2 x 2

80 = 2 x 2 x 2 x 2 x 5

The factors 2 x 2 x 2 x 2 appear common in both the cases. Hence the greatest common factor is 2 x 2 x 2 x 2 = 16.

A Venn diagram will be more clearer on this. The product of the numbers in the shaded area gives the greatest common factor.

 Related Calculators Calculate Greatest Common Factor Greatest Common Denominator Calculator Common Factor Calculator Least to Greatest

Applications of Greatest Common Factor

Greatest Common Factor are very useful in reducing fractions to the lowest terms and also in factoring expressions.

Example:

$\frac{32}{48}$ = $\frac{2 \times 16}{3 \times 16}$ = $\frac{2}{3}$   (The GCF of 32 and 48 is 16)

18ab2 – 27a3b = 9ab(2b – 3a2)  (The GCF of 18 and 27 is 9. The GCF of ab2 and a3b is ab)

Example Problems on Greatest Common Factor

Example 1:

Find the greatest common factor of 8 and 14.

Solution:

Step 1: Find the factors of 8

We can divide 8 by 1 or 2 or 4 or 8, so

Factors of 8 = 1, 2, 4, 8

Step 2: Find the factors of 14

We can divide 14 by 1 or 2 or 7 or 14, so

Factors of 14 = 1, 2, 7,14

Step 3: Find the common factors of 8 and 14

Common factors of 8 and 14 = 1, 2

Here, 2 is greater than 1. So, 2 is the greatest common factor

Step 4: Solution

Greatest common factor of 8 and 14 is 2

Example 2:

Find the greatest common factor of 9 and 27.

Solution:

Step 1: Find the factors of 9

We can divide 9 by 1 or 3 or 9, so

Factors of 9 = 1, 3, 9

Step 2: Find the factors of 27

We can divide 27 by 1 or 3 or 9 or 27, so

Factors of 27 = 1, 3, 9, 27

Step 3: Find the common factors of 9 and 27

Common factors of 9 and 27 = 1, 3, 9

Here, 9 is greater than 1 and 3. So, 9 is the greatest common factor.

Step 4: Solution

Greatest common factor of 9 and 27 is 9.

Practice Problems on Greatest Common Factor:

1) Find the greatest common factor of 6 and 16.

2) Find the greatest common factor of 18 and 24.

Solutions:

1) GCF of 6 and 16 is 2

2) GCF of 18 and 24 is 6

 More topics in  Greatest Common Factor How to Find the Greatest Common Factor Greatest Common Factor Word Problems Greatest Common Divisor
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