Finding the Factors of a Number: A Tutorial

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Learn how to factor with this simple guide
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A number’s factors are numbers that can be multiplied together to get the original number as a product or answer. In other words, every number is the product of multiple factors. Learning to factor or break up a number into its component factors is an important mathematical skill. We chatted with Advanced Math Teacher Taylor Klein to give you the best step-by-step instructions on how to factor small and big numbers, so keep reading to learn more!

Things You Should Know

  • Factoring a number is when you simplify the number into smaller products (or factors) of the number.
  • For example, 2 and 6 are factors of 12 because 2 × 6 equals 12.
  • The easiest way to factor a number is to try and divide it by the smallest prime number, such as 2 or 3.
Method 1
Method 1 of 2:

Factoring Basic Integers

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  1. To begin factoring, all you need is a number. [1] Any number will do, but for our purposes, let's start with a simple integer. Integers are numbers without fractional or decimal components (all positive and negative whole numbers are integers). [2]
    • For this example, let's use the number 12 .
  2. Advanced Math Teacher Taylor Klein says that “The first way you can factor a number is by finding its factor pairs.” Any integer can be written as the product or result of two other integers. Thinking of a number as the result of two factors can require "backwards" thinking. You essentially must ask yourself, “What multiplication problem equals this number?” [3]
    • In our example, 12 has multiple factors: 12 × 1, 6 × 2, and 3 × 4 all equal 12. So, we can say that 12's factors are 1, 2, 3, 4, 6, and 12 . Let’s work with the factors 6 and 2.
    • Even numbers are especially easy to factor because every even number has 2 as a factor. 4 = 2 × 2, 26 = 13 × 2, etc.
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  3. Lots of numbers—especially large ones—can be factored multiple times. When you've found two of a number's factors, if one has its own set of factors, you can also reduce this number to its factors. [4]
    • For instance, in our example, we have reduced 12 to 2 × 6. Notice that 6 has its own factors - 3 × 2 = 6. Thus, we can say that 12 = 2 × (3 × 2) .
  4. Prime numbers are numbers greater than 1 that are evenly divisible only by themselves and 1. For instance, 2, 3, 5, 7, 11, 13, and 17 are all prime numbers. Advanced Math Teacher Taylor Klein notes that when you've factored a number so it's the product of exclusively prime numbers, you can stop factoring.
    • In our example, we've reduced 12 to 2 × (2 × 3). 2, 2, and 3 are all prime numbers. If we were to factor further, we'd have to factor to (2 × 1) × ((2 × 1)(3 × 1)), which isn't typically useful, so it's usually avoided.
  5. Negative numbers can be factored nearly identically to how positive numbers are factored. The sole difference is that the factors must multiply together to make a negative number, so an odd number of the factors must be negative. [5]
    • For example, let's factor -60:
      • -60 = -10 × 6
      • -60 = (-5 × 2) × 6
      • -60 = (-5 × 2) × (3 × 2)
      • -60 = -5 × 2 × 3 × 2 . Note that having an odd number of negative numbers besides one will give the same product. For example, -5 × 2 × -3 × -2 also equals 60.
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Method 2
Method 2 of 2:

Factoring Larger Numbers

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  1. If you’re working with a larger number that has 4 or 5 digits, try using a table to organize your work. Write your number above a t-shaped table with two columns. [6]
    • For the purpose of our example, let's choose a 4-digit number to factor: 6,552 .
  2. Divide your number by the smallest prime factor (besides 1) that divides into it evenly with no remainder. Write the prime factor in the left column and write your answer across from it in the right column. As noted above, even numbers are easier to start factoring because their smallest prime factor will always be 2. Odd numbers, on the other hand, will have smallest prime factors that differ. [7]
    • In our example, since 6,552 is even, we know that 2 is its smallest prime factor. 6,552 ÷ 2 = 3,276. In the left column, we'll write 2 , and in the right column, write 3,276 .
  3. Repeat the previous step, but divide the newest number in the right column rather than dividing the number at the top of the table. Write the prime factor in the left column and the new number in the right column. Continue to repeat this process—with each repetition, the number in the right column should decrease. [8]
    • Let's continue with our process. 3,276 ÷ 2 = 1,638, so at the bottom of the left column, we'll write another 2 , and at the bottom of the right column, we'll write 1,638 . 1,638 ÷ 2 = 819, so we'll write 2 and 819 at the bottom of the two columns as before.
  4. It’s more difficult to find the smallest prime factor of odd numbers because they don't automatically have 2 as their smallest prime factor. When you reach an odd number, try dividing by small prime numbers other than 2—such as 3, 5, 7, 11, and so on—until you find one that divides evenly with no remainder. This is the number's smallest prime factor. [9]
    • In our example, we've reached 819. 819 is odd, so 2 is not a factor of 819. Instead of writing down another 2, we'll try the next prime number: 3. 819 ÷ 3 = 273 with no remainder, so we'll write down 3 and 273 .
    • When guessing factors, try all prime numbers up to the square root of the largest factor found so far. If none of the factors you try up to this point divide evenly, you're probably trying to factor a prime number and have solved the problem.
  5. Continue dividing the numbers in the right column by their smallest prime factor until you obtain a prime number in the right column. Divide this number by itself. This will put the number in the left column and "1" in the right column.
    • Let's finish factoring our number. See below for a detailed breakdown:
      • Divide by 3 again: 273 ÷ 3 = 91, no remainder, so we'll write down 3 and 91 .
      • Let's try 3 again: 91 doesn't have 3 as a factor, nor does it have the next lowest prime (5) as a factor, but 91 ÷ 7 = 13, with no remainder, so we'll write down 7 and 13 .
      • Let's try 7 again: 13 doesn't have 7 as a factor, or 11 (the next prime), but it does have itself as a factor: 13 ÷ 13 = 1. So, to finish our table, we'll write down 13 and 1 . We can finally stop factoring.
  6. Once you reach 1 in the right-hand column, you're done. The numbers listed on the left side of the table are your factors. In other words, when you multiply all these numbers together, the product will be the number at the top of the table. If the same factor appears multiple times, you can use exponent notation to save space. For instance, if your list of factors has four 2's, you can write 2 4 rather than 2 × 2 × 2 × 2. [10]
    • In our example 6,552 = 2 3 × 3 2 × 7 × 13 . This is the complete factorization of 6,552 into prime numbers. No matter what order these numbers are multiplied in, the product will be 6,552.
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Expert Q&A

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  • Question
    Can I multiply numbers with decimals?
    Taylor Klein
    Advanced Math Teacher
    Taylor Klein is an Advanced Math Teacher based in Philadelphia, Pennsylvania. She has worked in the education field for over 10 years, including eight years as a middle school Advanced Math Teacher. She has a master’s degree in Instructional Technology and Design and a master’s degree in Educational Leadership and Administration.
    Taylor Klein
    Advanced Math Teacher
    Expert Answer
    When multiplying numbers with decimals, follow the same multiplication process as you would with whole numbers, ensuring to account for the decimals after the calculation. The number of decimal places in your product should correspond to the total sum of decimal places in your factors.
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  • Question
    What number has 5 as a factor?
    Community Answer
    Any number ending in 0 or 5 would have 5 as a factor (e.g. 35, 450, or 764,545,230).
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    Is it necessary to arrange the factors of a number in ascending order?
    Community Answer
    It is customary but not mandatory, unless the instructions tell you to do so.
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      Tips

      • The lowest prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and 23.
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      • Some numbers can be factored in faster ways, but this method works every time, and, as an added bonus, the prime factors are listed in ascending order when you're done.
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      • Keep in mind that one number is a factor of another, larger number if it divides it cleanly. That is, the larger number can be divided by the smaller number without leaving a remainder. For instance, 6 is a factor of 24 because 24 ÷ 6 = 4 with no remainder.
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      Tips from our Readers

      The advice in this section is based on the lived experiences of wikiHow readers like you. If you have a helpful tip you’d like to share on wikiHow, please submit it in the field below.
      • Try using a divisibility test to find the smallest factor. For example, if you have a number like 114, you can add up all the digits: 1 + 1 + 4 = 6. 6 can be broken down into factors of 2 and 3. This means that 2 and 3 are also factors of 114.
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      Expert Interview

      Thanks for reading our article! If you’d like to learn more about math, check out our in-depth interview with Taylor Klein .

      References

      1. ↑ https://youtu.be/vcn2ruTOwFo?t=15
      2. ↑ https://www.cuemath.com/numbers/integers/
      3. ↑ https://sciencing.com/factors-number-quickly-easily-5192972.html
      4. ↑ https://sciencing.com/factors-number-quickly-easily-5192972.html
      5. ↑ https://sciencing.com/factor-negative-numbers-8170436.html
      6. ↑ https://sciencing.com/factors-number-quickly-easily-5192972.html
      7. ↑ https://sciencing.com/factors-number-quickly-easily-5192972.html
      8. ↑ https://youtu.be/AHOvHb3Ej_I?t=79
      9. ↑ http://amsi.org.au/teacher_modules/Multiples_factors_and_powers.html
      1. ↑ https://youtu.be/AHOvHb3Ej_I?t=100

      About This Article

      Article Summary X

      To factor a number, first find 2 numbers that multiply to make that number. For example, if you want to factor 12, you could use 4 and 3 since they multiply to make 12. Next, determine whether those 2 numbers can be factored again. In this example, 3 can't be factored again because it's a prime number, but 4 can be since 2 multiplied by 2 equals 4. Repeat this process until all of the numbers are prime numbers, and then write your answer like 12 = 2 × 2 × 3. If you want to learn strategies to help factor large numbers, keep reading the article!

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