Prime Factorization is a mathematical method employed to segregate a composite number into its prime factors. Composite numbers can be divided by more than just 1 and itself to conclude in a whole number. Prime numbers can just be divided by 1 and themselves, such as 2, 3, 5, and 7.
By working out the prime factors of a number, we could streamline equations and fractions, as well as learn the properties of that number. Let’s take a closer look.
Methods for Determining Prime Factorization
There are two main ways for performing prime factorization. You can use a factor tree, or you could factor by dividing by prime numbers. One as well as the other techniques will yield you the same result.
Factor Tree Method
The first method is using a factor tree. To make a factor tree:
Initiate by noting the number you are attempting to factor at the start of the page.
Underneath, create branches for the factors.
Proceed by creating branches and factoring until you arrive at a prime number.
Here’s an instance of the thought process:
We understand that 6 * 6 = 36.
Is 6 a prime number? No. 2 * 3 = 6.
Are 2 and 3 prime numbers? Yes. We have reached the end.
The prime factorization of 36 would be 2 x 2 x 3 x 3. To visualize this method, we could make a factor tree for 36 as follows.
As you can see, the prime factorization of 36 is figured out by multiplying the prime factors in the end of the tree, which are 2 * 3 * 2 * 3.
Division Method
The second process for working out the prime factorization of a number is to start using a record of prime numbers. Note down the value you are attempting to factor. Subsequently, divide the value by a prime number which divides evenly into it. It’s simplest to start with the initial few prime numbers, like 2, 3, and 5.
Let’s use the division process for prime factorization of 36.
What is a prime number which divides into 36? 2 → 2 * 18 = 36
Is 18 a prime number? No.
What is a prime number that divides into 18? 2 → 2 * 9 = 18
Is 9 a prime number? Nope.
What is a prime number that 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 depiction.
Using both process, we found the prime factorization for 36 is 2 * 2 * 3 * 3.
Examples of Prime Factorization
Although we’ve previously worked by the means of a few examples, let’s do one more employing both methods.
Let’s utilize a factor tree to work out the prime factorization of 100.
We know 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 test the division process:
What is the prime factorization for 100?
What is a prime number that 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 done here.
The prime factorization of 100 is 2 * 2 * 5 * 5.
Utilizations of Prime Factorization
Prime factorization has multiple practical utilizations in math. One of the major applications is simplifying fractions. By working out the prime factorization of the numerator and denominator of a fraction, you could cancel out common prime factors and simplify 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 clarified fraction of 1/3.
To see how this operates, let's employ prime factorization to streamline the fraction 8/24. First, perform prime factorization of the numerator and denominator.
8 = 2 * 2 * 2
24 = 2 * 2 * 2 * 3
Next, we rule out ever common prime factors among the numerator and denominator.
8/24 = 2 * 2 * 2 / 2 * 2 * 2 * 3 = 1/3
After canceling out the common prime factors of 2, we are left with the streamlined fraction of 1/3.
Prime factorization additionally executes an essential part in computer science and cryptography. As you proceed in sophisticated mathematics and science studies, you will figure out additional applications for prime factorization and be glad you are an expert.
It’s Prime Time to Boost Your Mathematical Abilities
Prime factorization is an important theory in mathematics that allows us to streamline equations and fractions, also learn the properties of a number. By practicing and understanding the processes and uses of prime factorization, we could enhance our math skills and analytical abilities.
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