## Prime Factorization: Overview

**Number theory** has fascinated mathematicians for years. Fundamental to number theory are numbers themselves, and the basic building blocks for numbers are **prime numbers**. A prime number is a counting number that has only two factors, itself and one. Counting numbers that have more than two factors (such as six, whose factors are 1, 2, 3, and 6), are said to be **composite numbers**. The number one has only one factor and is considered to be neither prime nor composite.

When a composite number is written as a product of all of its prime factors, we have the prime factorization of the number. For example, the number 36 can be written as a product of primes as: 36 = 2² x 3². The expression 2² x 3² is said to be the **prime factorization** of 36. The Fundamental Theorem of Arithmetic states that every composite number can be factored uniquely (except for the order of the factors) into a product of prime factors. What this means is that how you choose to factor a number into prime factors makes no difference. When you are done, the prime factorizations are essentially the same. Examine the two **factor trees** for 36 given below.

When we get done factoring using either set of factors to start with, we still have two factors of 2 and two factors of 3 or 2² x 3². This would be true if we had started to factor 36 as 12 times 3, 6 times 6, or any other pair of factors for 36.

Knowing the rules for divisibility will be very helpful when seeking to write a number in prime factorization form. Since a number is divisible by two if it ends in either 0, 2, 4, 6, or 8, it should be noted that two is the only even prime number. Another way to factor a number other than using factor trees is to start dividing by prime numbers, as shown below.

Once again, we can see that 36 = 2² x 3².
Another key idea in writing the prime factorization of a number is an understanding of **exponents**. An exponent tells how many times the base is used as a factor. In the prime factorization of 36 = 2² x 3², the two is used as a factor two times and the three is used as a factor twice.

When checking to see if a number is prime or not, you need only divide by those prime numbers that when squared remain less than the given number. For example, to see if 131 is prime, you need only check for divisibility by 2, 3, 5, 7, and 11, since 13² = 169. If a prime number greater than 13 divided 131, then the other factor would have to be less than 13 and you would have checked that factor already.