The method of prime factorization is used to “break down” or express a given number as product of prime numbers. More so, if a prime number occurs more than once in the factorization, it is usually expressed as exponential numbers to make it look more compact. Otherwise, we will have a long list of prime numbers being multiplied together. In this lesson, I have prepared eight (8) worked examples to explain the process of factorization using Prime numbers.
Let’s get started by defining first what a prime number is. Study both the examples and counterexamples of prime numbers carefully.
What is a Prime Number?
A prime number p is a whole number greater than 1 that is only divisible by 1 and itself. Another way of saying it, a prime number has exactly two factors, namely: 1 and itself.
Now, I will explain the general steps involved in performing prime factorization of a given positive integer.
Example 1: Find the prime factorization of 40 and express it in exponential notation.
I begin by listing the first few prime numbers in increasing order. The goal is to keep dividing the given number by an appropriate prime number starting from the lowest until the last quotient becomes prime as well.
There are two common ways to perform prime factorization. The first is called the Prime Factor Tree, and the second is known as the Upside-Down Division. With this, I will also show you two ways to prime factorize the number 40.
Example 2: Find the prime factorization of 32 and express it in exponential notation.
This is an even number, and thus divisible by prime number 2. So without hesitation, I begin using it as the starting divisor of choice.
Example 3: Find the prime factorization of 147 and express it in exponential notation.
I first recognize that 147 is an odd number, therefore, not divisible by 2. Move on to the next larger prime number, which in case 3.
I am not sure if you have encountered the “nice” the divisibility rule for number 3. It states that if the sum of the digits in a number is divisible by 3, then the original number is also divisible by 3.
We have 147, the sum of its digits is 147 = 1 + 4 + 7 = 12 which is divisible by 3. This implies that 147 must also be divisible by 3.
You don’t have to show every time the prime factorization of a number using the two methods. Just pick the one that is easy or convenient for you. For this exercise, I will utilize the Factor Tree method.
Example 4: Find the prime factorization of 540 and express it in exponential notation.
We know that any even number is always divisible by 2. So I would start dividing 540 by prime number 2. Let me use Upside-Down Division to prime factorize this number.
Example 5: Find the prime factorization of 945 and express it in exponential notation.
You may also do the prime factorization in its most straightforward way – that is, factor out the number horizontally and down the line. Just make sure that you always start with the lowest prime number, and go to the next larger one as needed to break it down until you end up with the final prime factor. Some textbooks do it this way to save space. It’s nice to add this method in your math “toolbox”.
It has five prime factors where three (3) of them are distinct, namely: 3, 5 and 7.
Example 6: Find the prime factorization of 1320 and express it in exponential notation.
Example 7: Find the prime factorization of 2025 and express it in exponential notation.
Hint: The given number is odd, so prime number 2 cannot divide it evenly. Start with 3.
Example 8: Find the prime factorization of 432 and express it in exponential notation.
Note: I used a slightly different method here. I call it, “Composite stays inside, Prime stays outside“. I am pertaining to the parenthesis in reference to where the composite or prime number stays.