A factor of a certain number, is another number that divides the certain number exactly, with no remainder.
For an example we can look at the number 15.
The numbers 1, 3, 5, 15 are the factors of 15.
Thus the factors of given numbers can be multiplied together, to result in the given number.
3 × 5 = 15 , 1 × 15 = 15
Common Factors
Common factors of multiple numbers, are factors that divide evenly into all of the numbers, and any pair or group of numbers could possibly have more than one common factor between them.Example
(1.1)
Factors of 12 => 1 , 2 , 3 , 4 , 6 , 12
Factors of 20 => 1 , 2 , 4 , 5 , 10 , 20
1 , 2 and 4 are common factors of 12 and 20.
Greatest Common Factor
The greatest common factor (GCF), is the largest common factor that a pair or group of numbers share in common with each other.Finding the GCF of two numbers is something that can be helpful for some situations such as simplifying fractions.
Example
(2.1)
Greatest common factor of 20 and 28?
Solution
In order to find the greatest common factor, we could list all the factors of each number, and look for the greatest shared factor.
Factors of 20 => 1 , 2 , 4 , 5 , 10 , 20
Factors of 28 => 1 , 2 , 4 , 7 , 14 , 28
From the lists it can be seen that 4 is the greatest common factor of 28 and 20.
Find GCF by Prime Factorization
In general though when wishing to find the greatest common factor of two numbers, rather than listing the factors out we can make use of prime factorization.
The Approach
Let’s look at the two numbers 12 and 56.The first thing to do is to prime factorize both numbers individually.
12
1) 12 ÷ 2 = 6
2) 6 ÷ 2 = 3 PRIME NUMBER
1) 12 ÷ 2 = 6
2) 6 ÷ 2 = 3 PRIME NUMBER
56
1) 56 ÷ 2 = 28
2) 28 ÷ 2 = 14
3) 14 ÷ 2 = 7 PRIME NUMBER
1) 56 ÷ 2 = 28
2) 28 ÷ 2 = 14
3) 14 ÷ 2 = 7 PRIME NUMBER
56 ) 2 × 2 × 2 × 7 = 56
When prime factorized, the numbers 12 and 56 both share one 2, and also another 2.
These prime factors that are shared, when multiplied together, will result in the greatest common factor (GCF), of the two numbers.
2 × 2 = 4 => The greatest common factor of 16 and 28 is 4.
This can also be seen by listing the factors of each number.
12 ) 1 , 2 , 3 , 4 , 6 , 12
56 ) 1 , 2 , 4 , 7 , 8 , 14 , 28 , 56
Example
(3.1)
Greatest common factor of 54 and 132?
Solution
54
1) 54 ÷ 2 = 27
2) 27 ÷ 2 = 13.5 Doesn’t divide exactly.
3) 27 ÷ 3 = 9
4) 9 ÷ 3 = 3 PRIME NUMBER
1) 54 ÷ 2 = 27
2) 27 ÷ 2 = 13.5 Doesn’t divide exactly.
3) 27 ÷ 3 = 9
4) 9 ÷ 3 = 3 PRIME NUMBER
132
1) 132 ÷ 2 = 66
2) 66 ÷ 2 = 33
3) 33 ÷ 2 = 16.5 Doesn’t divide exactly.
4) 33 ÷ 3 = 11 PRIME NUMBER
1) 132 ÷ 2 = 66
2) 66 ÷ 2 = 33
3) 33 ÷ 2 = 16.5 Doesn’t divide exactly.
4) 33 ÷ 3 = 11 PRIME NUMBER
132 ) 2 × 2 × 3 × 11 = 132
When prime factorized, the numbers 54 and 132 both share one 2, and one 3.
2 × 3 = 6 => The greatest common factor of 54 and 132 is 6.