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 conclude 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 understand the properties of that number. Let’s get a closer look.
Processes for Determining Prime Factorization
There are two main ways for working out prime factorization. You can employ a factor tree, or you could factor by dividing by prime numbers. Both techniques will give you the exact answer.
Factor Tree Method
The initial method is using a factor tree. To create a factor tree:
Start by noting the value you are trying to factor at the top of the page.
Under it, create branches for the factors.
Continue making branches and factoring before you reach a prime number.
Here’s an instance of the thought process:
We understand that 6 * 6 = 36.
Is 6 a prime number? No. 2 * 3 = 6.
Are 2 and 3 prime numbers? Yes. We are done here.
The prime factorization of 36 will be 2 x 2 x 3 x 3. To imagine this procedure, we could make a factor tree for 36 as ensues.
Consequently, 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 second way for working out the prime factorization of a number is to begin using a record of prime numbers. Write down the value you are attempting to factor. Then, divide the value by a prime number which divides evenly into it. It’s most convenient to begin with the first 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? No.
What is a prime number which divides into 18? 2 → 2 * 9 = 18
Is 9 a prime number? No.
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 visualization.
Utilizing both systems, we found the prime factorization for 36 is 2 * 2 * 3 * 3.
Examples of Prime Factorization
Despite the fact that we’ve previously worked through a few examples, let’s do another one using both methods.
Let’s utilize a factor tree to figure 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 clear.
The prime factorization of 100 is 2 * 2 * 5 * 5.
Let’s try the division process:
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 which divides into 50? 2 * 25 = 50
Is 25 a prime number? No.
What is a prime number which 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.
Utilizations of Prime Factorization
Prime factorization has many practical applications in mathematics. One of the primary utilizations 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 streamline the fraction.
For instance, the fraction 8/24 could be streamlined by solving 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 clarified fraction of 1/3.
To see how this works, let's utilize prime factorization to simplify the fraction 8/24. Primarily, perform prime factorization of the numerator and denominator.
8 = 2 * 2 * 2
24 = 2 * 2 * 2 * 3
Next, we cancel out any 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 remained with the streamlined fraction of 1/3.
Prime factorization additionally plays an essential role in cryptography and computer science. As you progress in sophisticated math and science theories, you will figure out further uses for prime factorization and be glad you are a pro.
It’s Prime Time to Enhance Your Math Abilities
Prime factorization is an important concept in math which enables us to streamline equations and fractions, as well as get a grasp of the characteristics of a value. By practicing and understanding the methods and utilizations of prime factorization, we can improve our math abilities and problem-solving abilities.
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