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The counting numbers (or natural numbers) are numbers with which we use to count things: 1, 2, 3, and so on. Counting numbers are generally divided into two categories: primes and composites. A prime number is a counting number that can only be divided by itself and 1. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, and 41. Of course this list goes on forever but generally you will not need prime numbers larger than these. The composite numbers are those counting numbers larger than 2 that are not primes. Notice that 1 is not in either list. One has its own special category called a unity.

A very beneficial rule in mathematics is the rule that says every composite number can be written as a product of prime numbers. This product is called prime factorization and is the subject of this webpage. For example, see below.

338 = 2 * 13 * 13

Prime factorization is useful for finding common denominators, greatest common factors, and least common multiples. But how do we take a large number and factor it into primes? We must have our list of prime numbers in front of us. Generally, we find the smallest prime number in the list that divides the given number. That will break the original number into two pieces, the prime we just found and another number. If the second number is not a prime, we find the smallest prime number that goes into it and break that second number into a product. We continue until all numbers are prime numbers.

Find the prime factorization of 2,223

2,223 = 2,223

2,223 = 3 * 741

2,223 = 3 * 3 * 247

2,223 = 3 * 3 * 13 * 19

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