Prime Factorization

In class today we worked on prime factorization.  Prime factorization is a factor string that contains only prime numbers.  See below:

If we start with 100, we can decompose 100 into 4 x 25 (decomposing is breaking a number into a known factor pair).  If the numbers are composite (meaning more than 2 factors), we can decompose those numbers further.  The prime factorization of 100 is 2 x 2 x 5 x 5 as you can see above.

There are two methods we can use to find the prime factorization of a number.  Method 1 is dividing.

As you can see, we divide using only prime numbers.  We start with 360 and divide by 2.  We end up with 180.  180 is divisible by 3, which leaves use 60.  60 divided by 3 gives us 20, then 20 divided 2 equals 10.  10 divided by 2 gives us 5, our last prime number.  We want to put our factor string in order from least to greatest so 2 x 2 x 2 x 3 x 3 x 5 is the prime factorization of 360.

The second method is the factor tree method.  You begin by writing 360, then using "branches" you decompose into two known factors.

We decomposed 360 into 36 x 10.  Obvious right?  Since 36 and 10 are composite, we can"branch" off of both of those.  36 can decompose to 6 x 6 and 10 can decompose to 5 x 2.  5 and 2 are prime numbers and therefore can no longer decompose.  6 is composite and decomposes to 3 x 2.  Prior to rearranging our factor string, we end up with 3 x 2 x 3 x 2 x 5 x 2.  We rearrange it to be in order from least to greatest leaving us with 2 x 2 x 2 x 3 x 3 x 5.

A step further would be to use exponents, which we did not cover in class yet.  We can rewrite our factor string by counting the number each time occurs and using an exponent to represent each number.  Since there are 3 2's and 2 3's we can rewrite the string to look like this.  (^ is a symbol used to denote exponents)

2^3 x 3^2 x 5