Q: What is the prime factorization of the number 31,404,560?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 5 x 11 x 127 x 281
    • or also written as { 2, 2, 2, 2, 5, 11, 127, 281 }
  • Written in exponential form: 24 x 51 x 111 x 1271 x 2811

Why is the prime factorization of 31,404,560 written as 24 x 51 x 111 x 1271 x 2811?

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 31,404,560

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 31,404,560 by 2

31,404,560 ÷ 2 = 15,702,280 - No remainder! 2 is one of the factors!
15,702,280 ÷ 2 = 7,851,140 - No remainder! 2 is one of the factors!
7,851,140 ÷ 2 = 3,925,570 - No remainder! 2 is one of the factors!
3,925,570 ÷ 2 = 1,962,785 - No remainder! 2 is one of the factors!
1,962,785 ÷ 2 = 981,392.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,962,785 ÷ 3 = 654,261.6667 - This has a remainder. 3 is not a factor.
1,962,785 ÷ 5 = 392,557 - No remainder! 5 is one of the factors!
392,557 ÷ 5 = 78,511.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
392,557 ÷ 7 = 56,079.5714 - This has a remainder. 7 is not a factor.
392,557 ÷ 11 = 35,687 - No remainder! 11 is one of the factors!
35,687 ÷ 11 = 3,244.2727 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
35,687 ÷ 13 = 2,745.1538 - This has a remainder. 13 is not a factor.
35,687 ÷ 17 = 2,099.2353 - This has a remainder. 17 is not a factor.
35,687 ÷ 19 = 1,878.2632 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
35,687 ÷ 127 = 281 - No remainder! 127 is one of the factors!
281 ÷ 127 = 2.2126 - There is a remainder. We can't divide by 127 evenly anymore. Let's try the next prime number
281 ÷ 131 = 2.145 - This has a remainder. 131 is not a factor.
281 ÷ 137 = 2.0511 - This has a remainder. 137 is not a factor.
281 ÷ 139 = 2.0216 - This has a remainder. 139 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
281 ÷ 281 = 1 - No remainder! 281 is one of the factors!

The orange divisor(s) above are the prime factors of the number 31,404,560. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 5 x 11 x 127 x 281 = 31,404,560. It can also be written in exponential form as 24 x 51 x 111 x 1271 x 2811.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 31,404,560.

31,404,560
Factor Arrows
215,702,280
Factor Arrows
27,851,140
Factor Arrows
23,925,570
Factor Arrows
21,962,785
Factor Arrows
5392,557
Factor Arrows
1135,687
Factor Arrows
127281

More Prime Factorization Examples

31,404,55831,404,55931,404,56131,404,562
21 x 31 x 1632 x 1971171 x 1,847,327131 x 1,0191 x 10,273121 x 71 x 2,243,1831

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