Prime Factorization is a mathematical function employed to split up 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 instance 2, 3, 5, and 7.
By figuring out the prime factors of a number, we can simplify equations and fractions, as well as learn the characteristics of that number. Let’s look closely.
Processes for Determining Prime Factorization
There are two main processes for working out prime factorization. You can employ a factor tree, or you can factor by dividing by prime numbers. Both processes will yield you the same result.
Factor Tree Method
The primary way is using a factor tree. To make a factor tree:
Initiate by noting the number you are trying to factor at the top of the page.
Below it, make branches for the factors.
Continue making branches and factoring before 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 have reached the end.
The prime factorization of 36 will be 2 x 2 x 3 x 3. To imagine this method, we can make a factor tree for 36 as ensues.
As you can see, the prime factorization of 36 is comprehended by multiplying the prime factors in the end of the tree, which are 2 * 3 * 2 * 3.
Division Method
The next method for figuring out the prime factorization of a number is to start using a record of prime numbers. Write down the number you are attempting to factor. Next, divide the number by a prime number that divides evenly into it. It’s easiest to start with the initial 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? 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.
The prime factorization of 36 is 2 * 2 * 3 * 3, and here’s a visualization.
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 through a few examples, let’s do one more utilizing both processes.
Let’s utilize a factor tree to work out 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 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? 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.
Uses of Prime Factorization
Prime factorization has multiple realistic utilizations in mathematics. One of the primary applications is simplifying 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 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 simplified fraction of 1/3.
To check how this functions, let's utilize prime factorization to simplify the fraction 8/24. First, 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 canceling out the common prime factors of 2, we are remained with the streamlined fraction of 1/3.
Prime factorization also plays an essential part in computer science and cryptography. As you progress in advanced mathematics and science concepts, you will figure out more utilizations for prime factorization and be pleased you are a professional.
It’s Peak Hour to Boost Your Mathematical Abilities
Prime factorization is an essential theory in mathematics which allows us to streamline equations and fractions, as well as learn the properties of a number. By practicing and understanding the processes and utilizations of prime factorization, we can enhance our math abilities and problem-solving abilities.
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