Factors

The factor of a whole number is any other whole number that can be divided into the first number without leaving a remainder. For example, 9 is a factor of 36 because 9 can be divided into 36 without a remainder: 36/9 = 4.

 

Factor Pairs

 

Factor pairs of a whole number are two factors that when multiplied together give the whole number. For example: 4 and 9 are factor pairs of the whole number 36, since 4 x 9 = 36.

 

Here are all the factor pairs of the number 36:

 

Factor Pairs

1,36

2,18

3,12

4,9

6,6

 

Number Models

1 * 36 = 36

2 * 18 = 36

3 * 12 = 36

4 * 9 = 36

6 * 6 = 36

 

 

The factors of 36 are: 1,2,3,4,6,9,12,18,36

 

Factor Trees

 

Let’s see what the factor tree of 14 looks like:

 

Start with 1 and 14 (1 x 14 = 14)

 

1                                                                             14

 

 

 

Now let’s see 2 and 7 (2 x 7 =14)

 

 

 

 


 

1       2                                                                   7    14

                                                                                  

The factor pairs of 14 are: 1 and 14, and 2 and 7

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Draw the factor tree for 16:

 

 

 


 

                                                                                        

                                

1       2       4                                                 4        8    16

 

 

The factor pairs of 16 are: 1 and 16,  2 and 8,  4 and 4

 

 

Prime Factors

 

The prime factors of a number are the factors of that number that are prime numbers:

 

Example: find the prime factors of 36:

 

Two factors of 36 are 6 x 6. But 6 is not a prime number. The factors of 6 are 2 x 3.

 

Therefore we can say the factors of 36 are: 2 x 3 x 2 x 3 = 36.

 

Putting the prime factors together: 2 x 2 x 3 x 3 = 36

 

We can also write this as: 22x 32 = 36

 

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The prime factors of 16 are:

 

2 x 8 = 2 x 4 x 2 = 2 x 2 x 2 x 2

 

Greatest Common Factor (GCF)

 

The GCF of two or more integers is the product of the prime factors common to all the integers.

 

Example: the GCF of 48 and 80 is:

 

40 = 2 · 2 · 2  · 5

80 = 2 · 2 · 2  2 · 2 · 2  · 5 2 · 5 

 

The common prime factors are: 2, 2, 2, and 5. So the GCF is: 2 · 2 · 2  · 5  = 40

 

Example:

 

36 x3 y2 = 2 · 2 · 3 · 3 · x · x · x · y · y

 

60 x2 y z = 2 ·  2 · 3 · 5 ·  x · x · y · z

 

The common prime factors are: 2, 2, 2, 3, x, x, and y The GCF is: 24 x2 y