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 result in a whole number. Prime numbers can just be divided by 1 and themselves, for example 2, 3, 5, and 7.
By finding the prime factors of a number, we can simplify fractions and equations, in addition understand the properties of that number. Let’s take a closer look.
Processes for Finding Prime Factorization
There are two primary methods for performing prime factorization. You could employ a factor tree, or you can factor by dividing by prime numbers. One as well as the other processes will yield you the same result.
Factor Tree Method
The initial method is utilizing a factor tree. To make a factor tree:
Start by noting the value you are trying to factor at the start of the page.
Underneath, make branches for the factors.
Continue making branches and factoring until you reach a prime number.
Here’s an instance of the thinking process:
We know 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 will be 2 x 2 x 3 x 3. To visualize this process, we could create a factor tree for 36 as follows.
As demonstrated, the prime factorization of 36 is figured out by multiplying the prime factors in the end of the tree, that is 2 * 3 * 2 * 3.
Division Method
The second method for working out the prime factorization of a number is to begin using a record of prime numbers. Write down the value you are trying to factor. Subsequently, divide the number by a prime number that divides evenly into it. It’s most convenient to start with the first few prime numbers, like 2, 3, and 5.
Let’s utilize 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 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 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 employing both processes.
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 try 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? No.
What is a prime number which divides into 50? 2 * 25 = 50
Is 25 a prime number? No.
What is a prime number that 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.
Utilizations of Prime Factorization
Prime factorization has several practical uses in mathematics. One of the primary uses is streamlining fractions. By finding out the prime factorization of the numerator and denominator of a fraction, you could cancel out common prime factors and streamline the fraction.
For instance, the fraction 8/24 could be simplified by solving the prime factorization of both values. 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 works, 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
Next, we rule out any common prime factors within 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 further plays an important role in computer science and cryptography. As you proceed in advanced mathematics and science theories, you will figure out further utilizations for prime factorization and be glad you are a professional.
It’s the Best Time to Advance Your Mathematical Skills
Prime factorization is an important concept in math which allows us to simplify equations and fractions, as well as understand the properties of a value. By practicing and comprehending the methods and utilizations of prime factorization, we can improve our mathematical skills and problem-solving skills.
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