Prime Factorization is a mathematical method used to break down a composite number into its prime factors. Composite numbers can be divided by more than just 1 and itself to result in a whole number. Prime numbers can just be divided by 1 and themselves, such as 2, 3, 5, and 7.
By finding the prime factors of a number, we can streamline fractions and equations, as well as understand the characteristics of that number. Let’s look closely.
Ways for Determining Prime Factorization
There are two main ways for working out prime factorization. You could utilize a factor tree, or you could factor by dividing by prime numbers. One as well as the other processes will yield you the same outcome.
Factor Tree Method
The primary way is using a factor tree. To create a factor tree:
Start by noting the number you are trying to factor at the top of the page.
Underneath, make branches for the factors.
Continue drawing branches and factoring before you reach a prime number.
Here’s an instance of the thinking process:
We recognize that 6 * 6 = 36.
Is 6 a prime number? No. 2 * 3 = 6.
Are 2 and 3 prime numbers? Yes. We are done.
The prime factorization of 36 will be 2 x 2 x 3 x 3. To visualize this process, we could make a factor tree for 36 as follows.
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 next method for working out the prime factorization of a number is to initiate utilizing a list of prime numbers. Write down the value you are trying to factor. Next, divide the number by a prime number which divides evenly into it. It’s easiest 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 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.
The prime factorization of 36 is 2 * 2 * 3 * 3, and here’s a chart.
Utilizing both process, we worked out the prime factorization for 36 is 2 * 2 * 3 * 3.
Examples of Prime Factorization
Despite the fact that we’ve already worked through a few examples, let’s do another one employing 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 clear.
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 which divides into 100? 2 * 50 = 100
Is 50 a prime number? No.
What is a prime number which 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.
The prime factorization of 100 is 2 * 2 * 5 * 5.
Applications of Prime Factorization
Prime factorization has multiple realistic uses 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 can cancel out common prime factors and simplify the fraction.
For example, the fraction 8/24 could be simplified by finding 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 functions, let's use prime factorization to simplify the fraction 8/24. Foremost, work on prime factorization of the numerator and denominator.
8 = 2 * 2 * 2
24 = 2 * 2 * 2 * 3
Subsequently, we rule out ever common prime factors within 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 computer science and cryptography. As you progress in advanced mathematics and science studies, you will find further utilizations for prime factorization and be delighted you are a pro.
It’s Peak Hour to Boost Your Math Abilities
Prime factorization is an essential concept in mathematics that enables us to streamline equations and fractions, as well as get a grasp of the properties of a value. By practicing and understanding the processes and uses of prime factorization, we can enhance our mathematical skills and problem-solving skills.
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