A factor of a number is any whole number that our starting number can be divided by, such that the answer is also a whole number. For example, 3 is a factor of 6 (6 divided by 3 equals 2), but 4 is not a factor of 6 (6 divided by 4 equals 1.5). Factors are useful tools in many areas of math, especially when dealing with fractions and ratios.
Finding the factors of a number can sometimes be tedious, especially for big numbers. Lets look at some tricks that can make this process easier.
1. 0 is Never a Factor
We are never allowed to divide by 0! if you think about it, how many times can you fit 0 into 5? 100? 1,000? 1,000,000? You can add up however many zeroes you want and you will never get any closer to 5. Therefore, 0 is never a factor of any number.
2. 1 is Always a Factor
We can always divide a number by 1. Any number divided by 1 is just itself. For example, 5 divided by 1 really means “what number fits into 5 once?” The answer would be 5 itself!
3. A Number is Always a Factor of Itself
This is the opposite of trick #2. If we take the example of 5 divided by 5, we are really asking “what number fits into 5 5 times?” The answer would be 1!
4. Factors Always Come in Pairs
You might have noticed from tricks #2 and #3 that all we did was flip the order of division (5 divided by 1 equals 5 and 5 divided by 5 equals 1). This is a hint that factors always come in pairs. The best way to think about this is by flipping the division problem to frame it as multiplication. Instead of asking “what numbers can I divide by?” we could ask “which 2 numbers multiply to get the desired number?” Taking our example of 5 again, we could ask “which 2 numbers multiply to get 5?” We would then notice that 1 x 5 =5, so one pair of factors is 1 and 5.
5. When 2 is a Factor
2 is a factor of a number when the last digit of a number is even (0, 2, 4, 6, or 8). For example, 2 is a factor of 4,316, but it is not a factor of 4,317.
6. When 3 is a Factor
A trick that can make determining whether 3 is a factor of a number easier is to add up the digits and see if 3 is a factor of the sum. For example, it is not obvious whether 3 is a factor of 4,317, but if we add up the digits: 4 + 3 + 1 + 7 = 15, we might remember that 5 x 3 = 15, so 3 is a factor of 4,317.
7. When 5 is a Factor
5 is a factor of a number if the last digit is 0 or 5.
8. When 10 is a Factor
10 is a factor of a number if the last digit is 0.
9. Find Factor’s in Pairs, Starting at 1 and Counting Up
Finding factors can seem like a random process. How do we know where to start? A good place to begin is to start at 1 (always a factor), find it’s pair, and then check the next number.
For example, we check if 1 is a factor and find it’s pair, then we check if 2 is a factor. If 2 is a factor, we find it’s pair then check if 3 is a factor. We repeat this process until 1 of 3 cases:
- We find a pair of factors that are the same number (i.e. 3 x 3 = 9)
- We find two factors that are consecutive (in order). For example, if we are finding the factors of 12, once we’ve found 3 x 4 = 12 we should have found all of the factors.
- We’ve tried all number up to half the value of the number we’re trying to find factors of. For example, if we’re trying to find the factors of 11, once we’ve checked whether 6 is a factor, we don’t need to continue.
Examples
Now that we’ve learned some tricks, let’s look at some examples.
- Find the factors of 12.
First, we can state the easiest pair of factors: 1 and 12 (1 x 12 = 12)
Next, is 2 a factor of 12? Yes! We know this because the last digit (2) is even. Now to find 2’s pair. 2 x 6 = 12, so the other factor is 6.
Now, is 3 a factor of 12? Again, yes! If we add up the digits, 1 + 2 =3, and 3 is a factor of 3. To find 3’s pair, we do 3 x 4 = 12, so the other factor is 4.
Have we found all of the factors? Yes! We found a pair of factors (3 and 4) that are consecutive, so we know we are done. The factors of 12 are 1, 2, 3, 4, 6, and 12!
2. Find the factors of 13
First, we can state the easiest pairs of factors: 1 and 13 (1 x 13 = 13)
Next, is 2 a factor of 13? No! The last digit (3) is odd, so 2 is not a factor.
Next, is 3 a factor of 13? No! If we add up the digits, 1 + 3 = 4. 3 is not a factor of 4.
We could then try 4, 5, 6, and 7 and we would find that these numbers can’t be evenly divided by 13. We know we are done, because 7 is the midpoint of 13. Therefore, the factors of 13 are 1 and 13. That’s it!
This has been an overview of tips to finding factors. For more information on this or other math topics as well as assistance with homework and test preparation, feel free to reach out to an Academic Director toll-free at 1 (877) 545-7737 or via our Contact Us page.
