Prime Factorization is a mathematical function 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 only be divided by 1 and themselves, such as 2, 3, 5, and 7.
By finding the prime factors of a number, we can simplify fractions and equations, as well as understand the properties of that number. Let’s take a closer look.
Methods for Determining Prime Factorization
There are two main methods for figuring out prime factorization. You could utilize a factor tree, or you could factor by dividing by prime numbers. Both techniques will grant you the exact outcome.
Factor Tree Method
The initial way is utilizing a factor tree. To make a factor tree:
Begin by writing the number you are attempting to factor at the start 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 envision this method, we can make a factor tree for 36 as follows.
As you can see, the prime factorization of 36 is figured out by multiplying the prime factors at the bottom of the tree, that is 2 * 3 * 2 * 3.
Division Method
The second method for figuring out the prime factorization of a number is to initiate using a list of prime numbers. Note the value you are attempting to factor. Subsequently, divide the value by a prime number which divides evenly into it. It’s simplest to initiate with the initial 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? Nope.
What is a prime number that 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.
Utilizing both systems, we found the prime factorization for 36 is 2 * 2 * 3 * 3.
Examples of Prime Factorization
Even though we’ve earlier worked through a few examples, let’s do another one utilizing both processes.
Let’s utilize 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 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? No.
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 here.
The prime factorization of 100 is 2 * 2 * 5 * 5.
Applications of Prime Factorization
Prime factorization has several practical uses in mathematics. One of the major uses is streamlining fractions. By finding out the prime factorization of the numerator and denominator of a fraction, you can cancel out common prime factors and simplify the fraction.
For instance, the fraction 8/24 could be streamlined 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 streamlined fraction of 1/3.
To see how this works, let's use 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
Subsequently, we cancel out ever 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 simplified fraction of 1/3.
Prime factorization also executes an essential part in computer science and cryptography. As you progress in advanced mathematics and science theories, you will figure out further uses for prime factorization and be pleased you are an expert.
It’s Prime Time to Boost Your Mathematical Skills
Prime factorization is an essential theory in math which enables us to simplify equations and fractions, as well as learn the characteristics of a number. By practicing and understanding the processes and applications of prime factorization, we can enhance our math abilities and problem-solving abilities.
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