Factor a Number

A number's factors are numbers which multiply together to form it as a product. Another way of thinking of this is that every number is the product of multiple factors. Learning how to factor - that is, break up a number into its component factors - is an important mathematical skill that is used not only in basic arithmetic but also in algebra, calculus, and beyond. See Step 1 below to start learning how to factor!

Steps

Factoring Basic Integers

  1. Write your number. To begin factoring, all you need is a number - 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).
    • Let's choose the number 12. Write this number down on a piece of scratch paper.
  2. Find two more numbers that multiply to make your first number. Any integer can be written as the product of two other integers. Even prime numbers can be written as the product of 1 and the number itself. Thinking of a number as the product of two factors can require "backwards" thinking - you essentially must ask yourself, "what multiplication problem equals this number?"
    • 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. For our purposes, 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.
  3. Determine whether any of your factors can be factored again. 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 reduce this number to its factors as well. Depending on the situation, it may or may not be beneficial to do this.
    • 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. Stop factoring when you reach prime numbers. 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. When you've factored a number so that it's the product of exclusively prime numbers, further factoring is superfluous. It does you no good to reduce each factor to itself times one, so you may stop.
    • 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. Factor negative numbers in the same way. 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 as their product, so an odd number of the factors must be negative.
    • For example, let's factor -60. See below:
      • -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.

Strategy for Factoring Large Numbers

  1. Write your number above a 2-column table. While it's usually fairly easy to factor small integers, larger numbers can be daunting. Most of us would be hard-pressed to break a 4 or 5-digit number into its prime factors using nothing but mental math. Luckily, using a table, the process becomes much easier. Write your number above a t-shaped table with two columns - you'll use this table to keep track of your growing list of factors.
    • For the purpose of our example, let's choose a 4-digit number to factor - 6,552.
  2. Divide your number by the smallest possible prime factor. 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 especially easy 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.
    • 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. Continue to factor in this fashion. Next, factor the number in the right column by its smallest prime factor, rather than 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.
    • Let's continue with our process. 3,276 ÷ 2 = 1,638, so att 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. Deal with odd numbers by trying small prime factors. Odd numbers are more difficult to find the smallest prime factor of than even 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 - 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.
    • 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, you should 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 thus are finished with the factoring process.
  5. Continue until you reach 1. 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. Use the numbers in the left-hand column as your original number's factors. 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, the product when you multiply all of these numbers together 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 24 rather than 2 × 2 × 2 × 2.
    • In our example 6,552 = 23 × 32 × 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.



Tips

  • 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.
  • 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
  • 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.
  • 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
  • If the numbers in the numerator add up to a multiple of three then three is a factor of that number. ( 819 = 8+1+9 which = 18, 1+8 =9. Three is a factor of nine so it is a factor of 819.)

Warnings

  • 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

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

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