Prime Factorization

This week was the very first week of Math 10 and we started out first unit. One of the first things we learned and in my opinion the most important thing was, prime factorization. Essentially prime factorization is the product of primes. Learning how to complete the prime factorization is a key step to solving many problems. We learned how to prime factorize a number in two different ways.

Method #1 – Factor Tree

(this is the easier method in my opinion)

Step 1 – starting the factor tree: At the top of the tree write the number you are trying to find the prime factorization for with two lines branching out from the bottom of the number. As an example lets use the number “220”. Start by finding two numbers that can multiply together to produce 220 and write them beneath the number.

Step 2 – continuing the factor tree: Next, repeat the first step until you get to the prime numbers (numbers that cannot be divided by anything other than 1 and itself) and cannot go anymore. Once you have gotten to this point you have prime factorized the number and found its prime factors. As you can see in the image below, at the bottom of the tree we have four numbers. 2, 2, 5, and 11 are all prime so we know the tree is finished.

Step 3 – writing out the prime factors: it is a very simple but necessary task to write out the prime factors of a number after completing the prime factorization. All that you will need to do is write down the amount of each number and write these numbers in a simple multiplication equation. Typically if there is more than one of a number exponents will be used.

Step 4 – finding the number of unique factors: After you have completed the prime factorization of the number you may need to find the specific number of unique factors (also known as prime factors) a number has. To do this simply add one number to the exponent of each number (numbers without an exponent have an exponent of 1). Then multiply these numbers together, your answer will be how many unique factors the number has.

Method #2 – Division Table

Step 1 – creating the division table: another method of completing the prime factorization of a number is to create a division table. Start by writing a number at the top of the table and checking if it is divisible by 2, the first prime number. If it is then write the quotient (the answer of the division equation) below. If it is not go onto the next prime number which is three, repeat this process until you get to a number that works.

Step 2 – continuing the process: all that is necessary now is to continue the process of finding prime numbers that the original number is divisible by and dividing them. Keep on going until the number on the right side is prime and it is not possible to go any further.

Note: one prime number can be used multiple times.

Step 3 – finding the prime factors: the prime factors in the division table are the numbers in the left column and the final prime number at the bottom of the table on the right side.

Step 4 – finding the number of unique factors: just like in the factor tree example above, in order to find how many unique factors a number has all you need to do is add one to the exponent of each number. After doing this multiply your resulting numbers together and then you should have the number of unique factors determined.

 

Summary

These are simple but effective methods to prime factorize numbers that can be used for many different problems such as finding the GCF or LCM of number. Once you learn this skill it makes many areas of math much simpler and easier to understand.