Prime Factorization is a mathematical method employed 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 working out the prime factors of a number, we can simplify equations and fractions, also understand the properties of that number. Let’s take a closer look.
Processes for Finding Prime Factorization
There are two main processes for figuring out prime factorization. You can use a factor tree, or you could factor by dividing by prime numbers. Both processes will grant you the same outcome.
Factor Tree Method
The initial way is utilizing a factor tree. To create a factor tree:
Initiate by noting the value you are attempting to factor at the top of the page.
Underneath, make branches for the factors.
Proceed by creating branches and factoring till you arrive at a prime number.
Here’s an example of the thinking 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 imagine this method, we could create a factor tree for 36 as follows.
As demonstrated, 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 begin using a list of prime numbers. Note the number you are attempting to factor. Subsequently, divide the number by a prime number that divides evenly into it. It’s simplest to initiate 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? Nope.
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 depiction.
Utilizing both methods, we found the prime factorization for 36 is 2 * 2 * 3 * 3.
Examples of Prime Factorization
Despite the fact that we’ve already worked by the means of a few examples, let’s do one more utilizing both methods.
Let’s utilize 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 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 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.
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 primary applications is simplifying fractions. By finding 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 could be simplified by finding 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 simplified fraction of 1/3.
To check how this functions, let's utilize prime factorization to simplify the fraction 8/24. Primarily, work on 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 remained with the streamlined fraction of 1/3.
Prime factorization additionally plays an important role in computer science and cryptography. As you proceed in progressive mathematics and science concepts, you will figure out additional applications for prime factorization and be glad you are a pro.
It’s Peak Hour to Boost Your Mathematical Abilities
Prime factorization is a crucial theory in math that enables us to simplify equations and fractions, as well as get a grasp of the characteristics of a value. By practicing and comprehending the methods and applications of prime factorization, we could improve our mathematical skills and problem-solving skills.
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