Q: What is the prime factorization of the number 225,232,452?

 A:
  • The prime factors are: 2 x 2 x 3 x 3 x 43 x 83 x 1,753
    • or also written as { 2, 2, 3, 3, 43, 83, 1,753 }
  • Written in exponential form: 22 x 32 x 431 x 831 x 1,7531

Why is the prime factorization of 225,232,452 written as 22 x 32 x 431 x 831 x 1,7531?

What is prime factorization?

Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.

Finding the prime factors of 225,232,452

To find the prime factors, you start by dividing the number by the first prime number, which is 2. If there is not a remainder, meaning you can divide evenly, then 2 is a factor of the number. Continue dividing by 2 until you cannot divide evenly anymore. Write down how many 2's you were able to divide by evenly. Now try dividing by the next prime factor, which is 3. The goal is to get to a quotient of 1.

If it doesn't make sense yet, let's try it...

Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Let's start by dividing 225,232,452 by 2

225,232,452 ÷ 2 = 112,616,226 - No remainder! 2 is one of the factors!
112,616,226 ÷ 2 = 56,308,113 - No remainder! 2 is one of the factors!
56,308,113 ÷ 2 = 28,154,056.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
56,308,113 ÷ 3 = 18,769,371 - No remainder! 3 is one of the factors!
18,769,371 ÷ 3 = 6,256,457 - No remainder! 3 is one of the factors!
6,256,457 ÷ 3 = 2,085,485.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
6,256,457 ÷ 5 = 1,251,291.4 - This has a remainder. 5 is not a factor.
6,256,457 ÷ 7 = 893,779.5714 - This has a remainder. 7 is not a factor.
6,256,457 ÷ 11 = 568,768.8182 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
6,256,457 ÷ 43 = 145,499 - No remainder! 43 is one of the factors!
145,499 ÷ 43 = 3,383.6977 - There is a remainder. We can't divide by 43 evenly anymore. Let's try the next prime number
145,499 ÷ 47 = 3,095.7234 - This has a remainder. 47 is not a factor.
145,499 ÷ 53 = 2,745.2642 - This has a remainder. 53 is not a factor.
145,499 ÷ 59 = 2,466.0847 - This has a remainder. 59 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
145,499 ÷ 83 = 1,753 - No remainder! 83 is one of the factors!
1,753 ÷ 83 = 21.1205 - There is a remainder. We can't divide by 83 evenly anymore. Let's try the next prime number
1,753 ÷ 89 = 19.6966 - This has a remainder. 89 is not a factor.
1,753 ÷ 97 = 18.0722 - This has a remainder. 97 is not a factor.
1,753 ÷ 101 = 17.3564 - This has a remainder. 101 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,753 ÷ 1,753 = 1 - No remainder! 1,753 is one of the factors!

The orange divisor(s) above are the prime factors of the number 225,232,452. If we put all of it together we have the factors 2 x 2 x 3 x 3 x 43 x 83 x 1,753 = 225,232,452. It can also be written in exponential form as 22 x 32 x 431 x 831 x 1,7531.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 225,232,452.

225,232,452
Factor Arrows
2112,616,226
Factor Arrows
256,308,113
Factor Arrows
318,769,371
Factor Arrows
36,256,457
Factor Arrows
43145,499
Factor Arrows
831,753

More Prime Factorization Examples

225,232,450225,232,451225,232,453225,232,454
21 x 52 x 4571 x 9,8571225,232,4511312 x 2231 x 1,051121 x 7,6491 x 14,7231

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