How to Factor a Number

Being able to factor numbers is a key skill, one that's necessary to learn and perform many other math skills down the line, like working with fractions. Fortunately, it's not as hard as it looks, and there's more than one correct way to do it. The technique listed below isn't always the fastest way, but it's the easiest to describe and follow.

Our process will be to find the smallest prime factor of our number (see the "Tips" section if you're not sure what "prime" means), divide our number by that prime, and repeat the process with the quotient until we reach 1. As an example, we'll illustrate the process by factoring the number 6,552. When we talk about "factoring" a number, we're talking about completely factoring it -- breaking it down into a list of prime factors. No matter what process you use, ultimately any counting number has only one complete factorization.

Steps   [edit]

  1. 6,552 completely factored. The numbers circled in red are the factors.
    Begin by writing the number (6552) on your paper. We'll be writing other numbers beneath it in two columns, so leave room for them. If you like, draw a vertical (up-and-down) line beneath the number to form the two columns.
  2. Start with the smallest prime number, which is 2. Is 2 a factor of 6,552? Yes it is, because 6,552 ÷ 2 = 3,276 with no remainder. In the left column, write 2, and in the right column, write 3276.
  3. Think about 3,276: does it have 2 as a factor? Yes, 3,276 ÷ 2 = 1,638, no remainder. At the bottom of the left column, write 2, and at the bottom of the right column, write 1638. As you see, 1,638 ÷ 2 = 819, had no remainder, so write 2 and 819 at the bottom of the two columns.
  4. Try dividing by 2 again: 819 ÷ 2 = 409 with a remainder of 1, so 2 is not a factor of 819. So instead of writing 2 down, we go to the next prime number: 3.
  5. Divide by 3: 819 ÷ 3 = 273, no remainder, so write down 3 and 273.
  6. Divide by 3 again: 273 ÷ 3 = 91, no remainder, so write down 3 and 91.
  7. 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 write down 7 and 13.
  8. 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 write down 13 and 1.
  9. Once you reach 1 in the right-hand column, you're done, and the numbers listed on the left are your factors: 6,552 = 2 × 2 × 2 × 3 × 3 × 7 × 13. That's the complete factorization of 6,552 into primes. Test it out, if you like: no matter what order you multiply those primes together, you wind up with 6,552.


Tips   [edit]

  • Remember that we're only talking about the so-called "natural numbers" -- sometimes called the "counting numbers": 1, 2, 3, 4, 5... We're not going to get into negative numbers or fractions, which might warrant their own articles
  • Understand 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. On the other hand, 6 is not a factor of 25
  • Also important is the concept of a prime number: a number that has only two factors, 1 and itself. 3 is a prime number because its only factors are 1 and 3. 4, on the other hand, has 2 as a factor. A number that isn't prime is called composite. (The number 1 itself, however, is considered neither prime nor composite -- it's a special case.)
  • The lowest prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and 23.
  • 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.
  • If your calculator has a square root button -- and it probably does -- you can sometimes save yourself effort by remembering this fact: the largest prime factor of a number (other than itself) can be no larger than the number's square root. For instance, if I'm looking for prime factors of 91, and my calculator tells me that the square root of 91 is about 9.54, then I know I don't have to look any higher than 9 for prime factors of 91. In fact, since 9 isn't prime, I don't have to look any higher than 7. (And 7 is a prime factor of 91, as we saw above.)


Warnings   [edit]

  • Don't make unnecessary work for yourself. Once you've eliminated a factor candidate, you don't have to test it again. Once we decided that 819 didn't have 2 as a factor, we didn't have to test 2 any further throughout the rest of the process.


Things You'll Need   [edit]

  • Paper
  • Writing tools, preferably pencil and eraser
  • Calculator (optional)