Prime Factorization is a mathematical method used to break down a composite number into its prime factors. Composite numbers could 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, for instance 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.
Methods for Determining Prime Factorization
There are two major methods for performing prime factorization. You can utilize a factor tree, or you could factor by dividing by prime numbers. Both methods will grant you the same outcome.
Factor Tree Method
The initial method is utilizing a factor tree. To create a factor tree:
Initiate by noting the value you are trying to factor at the top of the page.
Underneath, make branches for the factors.
Proceed by drawing branches and factoring until you arrive at 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 can create a factor tree for 36 as follows.
As demonstrated, the prime factorization of 36 is found by multiplying the prime factors at the bottom of the tree, which are 2 * 3 * 2 * 3.
Division Method
The next process for finding the prime factorization of a number is to begin using a record of prime numbers. Note the number you are attempting to factor. Subsequently, divide the number by a prime number which divides equally into it. It’s simplest to begin 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? Nope.
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.
Using both process, we established the prime factorization for 36 is 2 * 2 * 3 * 3.
Examples of Prime Factorization
Although we’ve earlier 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 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 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? No.
What is a prime number that 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.
Applications of Prime Factorization
Prime factorization has multiple real-life applications in math. One of the main utilizations is simplifying fractions. By finding 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 values. 8 = 2 * 2 * 2 and 24 = 2 * 2 * 2 * 3. By canceling out the common prime factor of 2, we are left with the simplified fraction of 1/3.
To check how this works, let's utilize prime factorization to simplify the fraction 8/24. First, perform prime factorization of the numerator and denominator.
8 = 2 * 2 * 2
24 = 2 * 2 * 2 * 3
Then, 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 left with the simplified fraction of 1/3.
Prime factorization also executes a crucial part in computer science and cryptography. As you proceed in sophisticated math and science theories, you will figure out more utilizations for prime factorization and be delighted you are an expert.
It’s Peak Hour to Advance Your Mathematical Abilities
Prime factorization is a crucial concept in mathematics which enables us to streamline equations and fractions, moreover understand the properties of a value. By practicing and understanding the processes and uses of prime factorization, we can enhance our math abilities and analytical abilities.
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