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
23 × 32 × 52 × 71
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.