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Prime Factors

Some numbers can be evenly divided only by 1 and themselves. These are prime numbers. Factors that are prime numbers are called prime factors.

Every whole number greater than one is either a prime number, or can be described as a product of prime factors.

Examples:10 is the product of the prime factors 2 x 5
11 is a prime number
12 is the product of the prime factors 2 x 2 x 3
324 is the product of the prime factors 2 x 2 x 3 x 3 x 3 x 3
700 is the product of the prime factors 2 x 2 x 5 x 5 x 7
701 is a prime number
2103 is the product of the prime factors 3 x 701

To find the prime factors of a given number, follow these steps:

  1. See if the number is a prime number. If it's below 1000, use the table of prime numbers. If it is prime, add it to the list of prime factors, and you're done.
  2. If it's not prime, try dividing it by a prime number, starting with 2.
  • If it divides cleanly, with no remainder, then add that prime to the list of prime factors. Take the quotient as your new number to work with, and return to step 1.
  • If it does not divide cleanly, return to step 2, but move on to the next prime on the list.
Example: Prime factors of 700700 ÷ 2 = 350, with no remainder. Add 2 to the list of prime factors.
350 ÷ 2 = 175, with no remainder. Add 2 to the list of prime factors.
175 ÷ 2 = 87.5. It doesn't divide cleanly, so we go to the next prime number.
175 ÷ 3 = 58.33. It doesn't divide cleanly, so we go to the next prime number.
175 ÷ 5 = 35, with no remainder. Add 5 to the list of prime factors.
35 ÷ 5 = 7, with no remainder. Add 5 to the list of prime factors.
7 is a prime number. Add 7 to the list of prime factors, and we're done.
The prime factors of 700 are 2 x 2 x 5 x 5 x 7.

Be sure to check at each step to see if the number you have is a prime. The next example illustrates why:

Example: Prime factors of 21032103 ÷ 2 is 1051.5. It doesn't divide cleanly, so we go to the next prime number.
2103 ÷ 3 is 701, with no remainder. Add 3 to the list of prime factors.
The table of prime numbers will tell you that 701 is a prime. Add 701 to the list of prime factors, and we're done.
The prime factors of 2103 are 3 x 701.

If we didn't notice that 701 was a prime, we'd have gone on to check 5, 7, 11, 13, and so on, going through 120 more primes before getting done. So be sure to check the quotient every time before proceeding.

(As you might imagine, this method is designed for smaller numbers. It's too time-consuming for very large ones.)


Information Please® Database, © 2007 Pearson Education, Inc. All rights reserved.

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