Q: What is the prime factorization of the number 451,016,232?

 A:
  • The prime factors are: 2 x 2 x 2 x 3 x 131 x 167 x 859
    • or also written as { 2, 2, 2, 3, 131, 167, 859 }
  • Written in exponential form: 23 x 31 x 1311 x 1671 x 8591

Why is the prime factorization of 451,016,232 written as 23 x 31 x 1311 x 1671 x 8591?

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 451,016,232

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 451,016,232 by 2

451,016,232 ÷ 2 = 225,508,116 - No remainder! 2 is one of the factors!
225,508,116 ÷ 2 = 112,754,058 - No remainder! 2 is one of the factors!
112,754,058 ÷ 2 = 56,377,029 - No remainder! 2 is one of the factors!
56,377,029 ÷ 2 = 28,188,514.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
56,377,029 ÷ 3 = 18,792,343 - No remainder! 3 is one of the factors!
18,792,343 ÷ 3 = 6,264,114.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
18,792,343 ÷ 5 = 3,758,468.6 - This has a remainder. 5 is not a factor.
18,792,343 ÷ 7 = 2,684,620.4286 - This has a remainder. 7 is not a factor.
18,792,343 ÷ 11 = 1,708,394.8182 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
18,792,343 ÷ 131 = 143,453 - No remainder! 131 is one of the factors!
143,453 ÷ 131 = 1,095.0611 - There is a remainder. We can't divide by 131 evenly anymore. Let's try the next prime number
143,453 ÷ 137 = 1,047.1022 - This has a remainder. 137 is not a factor.
143,453 ÷ 139 = 1,032.036 - This has a remainder. 139 is not a factor.
143,453 ÷ 149 = 962.7718 - This has a remainder. 149 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
143,453 ÷ 167 = 859 - No remainder! 167 is one of the factors!
859 ÷ 167 = 5.1437 - There is a remainder. We can't divide by 167 evenly anymore. Let's try the next prime number
859 ÷ 173 = 4.9653 - This has a remainder. 173 is not a factor.
859 ÷ 179 = 4.7989 - This has a remainder. 179 is not a factor.
859 ÷ 181 = 4.7459 - This has a remainder. 181 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
859 ÷ 859 = 1 - No remainder! 859 is one of the factors!

The orange divisor(s) above are the prime factors of the number 451,016,232. If we put all of it together we have the factors 2 x 2 x 2 x 3 x 131 x 167 x 859 = 451,016,232. It can also be written in exponential form as 23 x 31 x 1311 x 1671 x 8591.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 451,016,232.

451,016,232
Factor Arrows
2225,508,116
Factor Arrows
2112,754,058
Factor Arrows
256,377,029
Factor Arrows
318,792,343
Factor Arrows
131143,453
Factor Arrows
167859

More Prime Factorization Examples

451,016,230451,016,231451,016,233451,016,234
21 x 51 x 71 x 471 x 137,08715931 x 760,5671451,016,233121 x 225,508,1171

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