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 only be divided by 1 and themselves, for example 2, 3, 5, and 7.
By finding the prime factors of a number, we can streamline fractions and equations, in addition understand the properties of that number. Let’s get a closer look.
Ways for Finding Prime Factorization
There are two main ways for performing prime factorization. You can use a factor tree, or you can factor by dividing by prime numbers. One as well as the other techniques will give you the same answer.
Factor Tree Method
The first method is utilizing a factor tree. To create a factor tree:
Start by writing the number you are attempting to factor at the top of the page.
Below it, make branches for the factors.
Continue making branches and factoring till you reach a prime number.
Here’s an instance 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.
The prime factorization of 36 will be 2 x 2 x 3 x 3. To imagine this procedure, we could create a factor tree for 36 as ensues.
As demonstrated, the prime factorization of 36 is found by multiplying the prime factors at the bottom of the tree, that is 2 * 3 * 2 * 3.
Division Method
The second method for finding the prime factorization of a number is to initiate utilizing a list of prime numbers. Write down the number you are trying to factor. Then, divide the number by a prime number which divides evenly into it. It’s most convenient to begin 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 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? 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.
Using 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 previously worked through a some examples, let’s do one more using both processes.
Let’s use a factor tree to figure 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 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? Nope.
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 could cancel out common prime factors and clarify the fraction.
For instance, the fraction 8/24 can be simplified by solving the prime factorization of both numbers. 8 = 2 * 2 * 2 and 24 = 2 * 2 * 2 * 3. By canceling out the common prime factor of 2, we are left with the streamlined fraction of 1/3.
To check how this operates, let's employ 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
Next, we cancel 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 remained with the streamlined fraction of 1/3.
Prime factorization also plays an essential role in computer science and cryptography. As you proceed in advanced math and science theories, you will find additional uses for prime factorization and be delighted you are an expert.
It’s Prime Time to Advance Your Mathematical Abilities
Prime factorization is an essential theory in mathematics which enables us to simplify equations and fractions, as well as learn the characteristics of a number. By practicing and understanding the methods and applications of prime factorization, we can improve our math abilities and problem-solving abilities.
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