Prime Factorization is a mathematical method utilized 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 instance 2, 3, 5, and 7.
By figuring out the prime factors of a number, we can simplify fractions and equations, in addition understand the characteristics of that number. Let’s take a closer look.
Processes for Finding Prime Factorization
There are two primary methods for working out prime factorization. You could use a factor tree, or you could factor by dividing by prime numbers. One as well as the other techniques will grant you the exact outcome.
Factor Tree Method
The initial method is utilizing a factor tree. To create a factor tree:
Start by writing the number you are attempting to factor at the top of the page.
Underneath, make branches for the factors.
Proceed by making branches and factoring before you attain a prime number.
Here’s an instance 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 visualize this procedure, we can create a factor tree for 36 as follows.
As demonstrated, the prime factorization of 36 is comprehended by multiplying the prime factors at the bottom of the tree, that is 2 * 3 * 2 * 3.
Division Method
The next method for figuring out the prime factorization of a number is to initiate using a list of prime numbers. Note the value you are trying 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 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 that 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.
The prime factorization of 36 is 2 * 2 * 3 * 3, and here’s a depiction.
Utilizing both systems, we established 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 some examples, let’s do another one utilizing both methods.
Let’s use 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 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 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 clear.
The prime factorization of 100 is 2 * 2 * 5 * 5.
Uses of Prime Factorization
Prime factorization has multiple practical utilizations in mathematics. One of the main applications is streamlining fractions. By finding 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 canceling out the common prime factor of 2, we are left with the simplified fraction of 1/3.
To check how this operates, let's use 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 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 left with the simplified fraction of 1/3.
Prime factorization further executes an important role in cryptography and computer science. As you go ahead in advanced mathematics and science theories, you will find additional applications for prime factorization and be delighted you are a pro.
It’s the Best Time to Advance Your Math Abilities
Prime factorization is a crucial concept in mathematics that enables us to simplify equations and fractions, as well as understand the characteristics of a number. By practicing and understanding the methods and utilizations of prime factorization, we can improve our math abilities and analytical abilities.
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