Prime Factorization

Factors are the numbers you multiply to get another number.  For example, the factors of 15 are 1, 3, 5 and 15 because 1×15 = 15 and 3×5 = 15.  Most numbers have more than one factorization, for instance, 20 can be factored as 1×20, 2×10, or 4×5.  A prime number is one that can only be factored as 1 times itself.  The first few primes are 2, 3, 5, 7, 11, 13, 17 and 19.  The number 1 is not regarded as a prime and is usually not included in prime factorizations.

In prime factorization, you want to find the list of all prime-number factors of a given number.  For example, the prime factorization of 12 is 2×2×3 (usually written as 2,2,3).  Although 4 and 6 are factors of 12, they are not prime numbers and therefore not included in the prime factorization.

Prime factorization is often best since it avoids counting any factor too many times.  Suppose you need to find the prime factorization of 30.  Sometimes a student will list all the factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30…don’t make that mistake or your product will wind up as 810,000!  It’s best to stick with prime factorization, even if the problem doesn’t require it, to avoid omitting a factor or over-duplicating one.

In the case of 30 you can find the prime factorization by taking 30 and dividing it by a prime number that goes into 30 evenly: 30÷3 = 10.  If your quotient is not prime, divide it again by a prime number that will go in evenly: 10÷5 = 2.  When your last quotient is a prime number, you’e done!  So, for 30, the prime factors are 2,3,5 (note that the preferred order for listing them is from least to greatest).
An easy way of computing the factorization is to do upside-down division.  It could look like this:

You may have noticed that the computations in the written example are in a different order than the graphic example.  It really makes no difference as long as the divisors are prime numbers, they go into the dividend evenly, and the final quotient is a prime number…just remember to list them from least to greatest.  The nice thing about this method is, when you’re done, the prime factorization is the product of all the divisors plus the final quotient.

Here’s another example.
Find the prime factorization of 700
Do the upside-down division:
And the answer is:  700 = 2,2,5,5,7
Some teachers may prefer that the answer be written using exponential notation and the final answer would be written 2²,5²,7.

By the way, there are some divisibility rules that you can use to help you find the divisors.  There are several, but the simplest to use are these:

  • If the number is even, then it’s divisible by 2.
  • If the sum of the number’s digits is divisible by 3 then the number itself is divisible by 3.
  • If the number ends with a 0 or a 5 then it’s divisible by 5.

If you run out of small primes and you still have a large number, try bigger primes, 11, 13, 17, 19, etc. until you find something that works or until you reach primes whose squares are bigger than the number into which you’re trying to divide.  If the square of the prime you’re trying is bigger than the number, then the number left must be prime, and you’re done.

Here’s a tough example:

Find the prime factors of 3219

Since the sum of the numbers can be divided by three, we know one of the factors is 3 and 3219 ÷ 3 = 1073…now what?  Well, that’s why calculators were invented and some will calculate prime factors.  Even if yours doesn’t, it’s fairly easy to divide by increasingly large primes until you find one that works.  But we’ll be kind and help you out: Prime Factors Calculator.  You can also find this calculator on the Worksheets page.

Leave a Reply

Your email address will not be published. Required fields are marked *