Perhaps one of the most important and fundamental concepts when learning Number Theory is Prime Factorization. Prime numbers are the building blocks of Number Theory because all of the Natural numbers can be expressed as a unique product of prime factors. This is what is called Prime Factorization. Knowing the prime factors of a number also gives us insight into other properties of that number as we shall see when working with Least Common Multiples (LCM) and Greatest Common Factors (GCF) in the following chapters.
A formal definition of Prime Factorization is the decomposition of a composite number into a product of prime numbers. In other words, we can ask ourselves, “which prime numbers can we multiply together to get our original number?”
Example: What is the Prime Factorization of 72?
- We can start finding the prime factors by dividing our original number (72) by the first prime number (2). 72/2 = 36.
- We continue dividing by 2 until we no longer can and move on to the next prime (3) if necessary repeating this process until our dividend is a prime number. 36/2 = 18, 18/2 = 9, 9/3 = 3.
- Putting it all together we see the prime factors of 72 are 2, 2, 2, 3, 3. Another way to express this is: 23 * 32.
Example: What are the prime factors of 31?
- 31 is prime so it does not have any prime factors other than itself.