How to Find How Many Factors Are in a Number


In math, every number has factors. Factors can divide evenly into a number with no remainder or decimal. For example, 30 divided by 10 is 3, and 30 divided by 15 is 2, so 2, 3, 10, and 15 are all factors of 30. There is an easy process to find the factors of any number with a pencil and paper. You can also use a calculator for an easy shortcut.

Part 1
Constructing a Factor List

Constructing a Factor List on How to Find How Many Factors Are in a Number

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All whole numbers have 1 and themselves as factors, so your first two factors are 1 and the original number. Space these original two numbers apart because as you proceed, you’ll fill more factors in the middle.[1] If your original number is 30, then write 1 on the left side and 30 on the right side. As you find more factors, write the low factors on the left side and the high factors on the right side. This gradually fills in the space you left between the two numbers.

Constructing a Factor List on How to Find How Many Factors Are in a Number

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Work your way up from 1 and use long division to see if each number divides into the original number. For example, start with 2 and see it divides into 30. 30/2 = 15, so 2 and 15 are both factors of 30. Add these to your factor list to get 1, 2 … 15, 30.[2] Factors must be whole numbers. If you divide and get a decimal or a remainder, then this number is not a factor. For example, 30/4 is 7.5. This is not a whole number, so don’t include it. Remember that both the number you divide into the original number and the result are factors. For example, since 30/2 = 15, both 2 and 15 are factors of 30. Fill these numbers into your factor list. All the positive factors for 30 are 1, 2, 3, 5, 6, 10, 15, 30. Keep this list in mind as you move on.

Constructing a Factor List on How to Find How Many Factors Are in a Number

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At some point, your factor list will close and there won’t be any more whole numbers left that divide into the original number.[3] If you work your way inward, the last 2 factors you find will be 5 and 6. Since 30/5 = 6, and there are no whole numbers in between them that divide into 30, you've found the last 2 factors for 30. In other cases, there might still be whole numbers between your last 2 factors, but they won’t divide into the original number. Check by using long division. If there is a remainder or a decimal, then this number is not a factor.

Constructing a Factor List on How to Find How Many Factors Are in a Number

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To confirm your work, work your way inward from each end of the line and multiply each number by its opposite on the other side.[4] For example, the factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. In this case, 1 x 30 = 30, 2 x 15 = 30, 3 x 10 = 30, and so on. If you multiply 2 numbers and they don't give you 30, then double-check your work. You may have found an incorrect factor.

Constructing a Factor List on How to Find How Many Factors Are in a Number

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Remember that in math, a negative number multiplied by another negative number produces a positive number. This means that each factor you found can be negated and still produce the same result. Simply re-write your list of whole factors and add a negative sign to each number.[5] Including negative numbers, your whole factor list should be 1, 2, 3, 5, 6, 10, 15, 30, -1, -2, -3, -5, -6, -10, -15, -30. Remember that in math, multiplying 2 negative numbers or 2 positive numbers produces a positive result. But multiplying a negative by a positive produces a negative result. For example, 2 x -15 produces -30, so these numbers multiplied together aren’t factors of 30. However, -2 x -15 = 30, so these numbers are factors.

Part 2
Using a Graphing Calculator

Using a Graphing Calculator on How to Find How Many Factors Are in a Number

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These types of calculators can perform many advanced functions, including automatically generating all the factors of a number. With the correct keystrokes, you can use this easy shortcut for finding a number’s factors.[6] If you or your parents took math in high school or college, you may still have one of these graphing calculators laying around. There are also websites that have automatic factor calculators. On these sites, you just type in any number and the site will automatically give you all its factors.

Using a Graphing Calculator on How to Find How Many Factors Are in a Number

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Depending on your calculator model, this button should be just below the screen on the left side of the calculator. It will open a menu that shows a series of “Y=” options.[7]Clear any numbers that are in the Y= menu before entering your own equation.

Using a Graphing Calculator on How to Find How Many Factors Are in a Number

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” This programs the calculator to produce a table showing all the factors of your original number. [8]If your original number is 30, then enter 30/X. The full equation is Y1=30/X.

Using a Graphing Calculator on How to Find How Many Factors Are in a Number

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The 2nd key is usually a yellow button on the left side of the keyboard. This prompts the calculator to use a key’s alternate function when you press it. The alternate function of the Graph key is Table, so this produces a table of factors for your original number. [9]

Using a Graphing Calculator on How to Find How Many Factors Are in a Number

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This equation produces a table that shows all the factors of the original number. If 1 is in the X column, then 30 will be in the Y column. Then you can scroll to see how many factors appear for your number.[10] Stop scrolling when your original number is in the X column. When you reach 30 in the X column, the Y column will be 1. Stop here, as you’ve gone through all your factors. The table might not be at 1 by default. If not, use the arrow keys to scroll down until you reach 1.

Using a Graphing Calculator on How to Find How Many Factors Are in a Number

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Your calculator might include all numbers between 1 and the original number, even ones that don't divide evenly. You can easily tell which numbers don’t belong because they have a decimal place. Any number pairs that include a decimal place are not factors.[11]

Using a Graphing Calculator on How to Find How Many Factors Are in a Number

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In math, negative numbers are factors too because a negative number multiplied by another negative number produces a positive number. Your calculator might not display this, but you can easily negate the numbers by hand.[12] Simply write down the list of factors your calculator produces. Then add a negative sign at the beginning of each one to get all the negative factors. Including negative numbers, your whole factor list would be 1, 2, 3, 5, 6, 10, 15, 30, -1, -2, -3, -5, -6, -10, -15, -30. Remember that only 2 negative numbers produce a positive result in multiplication. For instance, 2 x -15 will produce -30, so these numbers multiplied together aren’t factors of 30. However, -2 x -15 = 30, so these numbers are factors.