Counting factors
Counting factors is not a bad idea when you are looking for the factors of a number.
You may need this information if you do not want to miss any factors.
To count factors, first we need to get the prime factorization of the number
Example #1:
How many factors does 8 have?
8 = 2 × 2 × 2 = 2
^{3}
The factors of 8 are 1, 2, 4, 8. There are 4 factors
Looking at 2
^{3}, we notice that if we add 1 to the exponent, we get the 4
However, just one case is not enough to conclude that when counting factors the number of factors is whatever the exponent is plus 1
Let's look at more examples
Example #2:
How many factors does 25 have?
25 = 5 × 5 = 5
^{2}
The factors of 25 are 1, 5, and 25. 25 has 3 factors
Again, to get the 3, just add 1 to the exponent of 2
Example #3:
How many factors does 72 have?
72 = 8 × 9 = 2 × 2 × 2 × 3 × 3 = 2
^{3} × 3
^{2}
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. There are 12 factors
How do we get the 12? By adding 1 to each exponent and then multiply
(3 + 1) × (2 + 1) = 4 × 3 = 12
So far it seems like adding 1 is a good strategy when counting factors
Another way to see all the factors of 72 are shown below:
2
^{0} × 3
^{0} = 1 × 1 = 1
2
^{0} × 3
^{1} = 1 × 3 = 3
2
^{0} × 3
^{2} = 1 × 9 = 9
2
^{1} × 3
^{0} = 2 × 1 = 2
2
^{1} × 3
^{1} = 2 × 3 = 6
2
^{1} × 3
^{2} = 2 × 9 = 18
2
^{2} × 3
^{0} = 4 × 1 = 4
2
^{2} × 3
^{1} = 4 × 3 = 12
2
^{2} × 3
^{2} = 4 × 9 = 36
2
^{3} × 3
^{0} = 8 × 1 = 8
2
^{3} × 3
^{1} = 8 × 3 = 24
2
^{3} × 3
^{2} = 8 × 9 = 72
As you can see there are 4 choices for the exponents of 2: 0, 1, 2, 3.
And 4 choices for the exponents of 3: 0, 1, 2
4 choices × 3 choices = 12 choices and this is equal to 12 factors
Example #4:
How many factors 12600 have?
When counting factors for big numbers, it may be useful to make a factor tree
Pull out all the prime numbers from the tree and multiply the numbers. This is your prime factorization.
2 × 2 × 2 × 3 × 3 × 5 × 5 × 7
2
^{3} × 3
^{2} × 5
^{2} × 7
^{1}
Add 1 to each exponent and multiply:
(3 + 1) × (2 + 1) × (2 + 1) × (1 + 1)
4 × 3 × 3 × 2
12 × 3 × 2
36 × 2 = 72
12600 has 72 factors.
Take the counting factors quiz below to see how well you understand this lesson.

Nov 09, 18 09:40 AM
Learn the three properties of congruence. Examples to illustrate which property.
Read More
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.