Prime Factorization is a mathematical function utilized to split up a composite number into its prime factors. Composite numbers could be divided by more than just 1 and itself to conclude in a whole number. Prime numbers can only be divided by 1 and themselves, for example 2, 3, 5, and 7.
By figuring out the prime factors of a number, we can simplify fractions and equations, also understand the properties of that number. Let’s get a closer look.
Processes for Finding Prime Factorization
There are two primary methods for figuring out prime factorization. You could employ a factor tree, or you could factor by dividing by prime numbers. Both methods will give you the exact result.
Factor Tree Method
The initial method is using a factor tree. To make a factor tree:
Start by writing the number you are trying to factor at the top of the page.
Below it, make branches for the factors.
Proceed by drawing branches and factoring before you arrive at a prime number.
Here’s an example of the thought process:
We know that 6 * 6 = 36.
Is 6 a prime number? No. 2 * 3 = 6.
Are 2 and 3 prime numbers? Yes. We are done here.
The prime factorization of 36 will be 2 x 2 x 3 x 3. To imagine this process, we could create a factor tree for 36 as ensues.
As you can see, the prime factorization of 36 is found by multiplying the prime factors in the end of the tree, that is 2 * 3 * 2 * 3.
Division Method
The second process for working out the prime factorization of a number is to initiate using a record of prime numbers. Write down the value you are attempting to factor. Next, divide the value by a prime number that divides equally into it. It’s simplest to start with the first few prime numbers, like 2, 3, and 5.
Let’s use the division method for prime factorization of 36.
What is a prime number that divides into 36? 2 → 2 * 18 = 36
Is 18 a prime number? No.
What is a prime number which divides into 18? 2 → 2 * 9 = 18
Is 9 a prime number? No.
What is a prime number which divides into 9? 3 → 3 * 3 = 9
Is 3 a prime number? Yes. We are done here.
The prime factorization of 36 is 2 * 2 * 3 * 3, and here’s a visualization.
Utilizing both systems, we established the prime factorization for 36 is 2 * 2 * 3 * 3.
Examples of Prime Factorization
Although we’ve already worked through a few examples, let’s do one more using both methods.
Let’s use a factor tree to find the prime factorization of 100.
We learned that 4 * 25 = 100.
2 * 2 = 4, and 2 is a prime number. 5 * 5 = 25, and 5 is a prime number. We are done.
The prime factorization of 100 is 2 * 2 * 5 * 5.
Let’s try the division method:
What is the prime factorization for 100?
What is a prime number which divides into 100? 2 * 50 = 100
Is 50 a prime number? Nope.
What is a prime number that divides into 50? 2 * 25 = 50
Is 25 a prime number? Nope.
What is a prime number which divides into 25? 5 * 5 = 25
Is 5 a prime number? Yes. We are clear.
The prime factorization of 100 is 2 * 2 * 5 * 5.
Uses of Prime Factorization
Prime factorization has multiple realistic uses in mathematics. One of the main applications is simplifying fractions. By finding the prime factorization of the numerator and denominator of a fraction, you can cancel out common prime factors and streamline the fraction.
For example, the fraction 8/24 could be simplified by working out the prime factorization of both numbers. 8 = 2 * 2 * 2 and 24 = 2 * 2 * 2 * 3. By ruling out the common prime factor of 2, we are left with the simplified fraction of 1/3.
To check how this operates, let's utilize prime factorization to simplify the fraction 8/24. Primarily, perform prime factorization of the numerator and denominator.
8 = 2 * 2 * 2
24 = 2 * 2 * 2 * 3
Then, we cancel out any common prime factors between the numerator and denominator.
8/24 = 2 * 2 * 2 / 2 * 2 * 2 * 3 = 1/3
After ruling out the common prime factors of 2, we are left with the streamlined fraction of 1/3.
Prime factorization further executes an essential role in cryptography and computer science. As you proceed in sophisticated mathematics and science theories, you will figure out more uses for prime factorization and be pleased you are a pro.
It’s Peak Hour to Advance Your Math Skills
Prime factorization is an important theory in math which allows us to simplify equations and fractions, also learn the properties of a value. By practicing and understanding the processes and uses of prime factorization, we can improve our mathematical abilities and analytical skills.
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With the assistance of Grade Potential tutoring, you could advance your math skills, improve your confidence, and attain your academic aims.
