Q: What is the prime factorization of the number 31,001,302?

 A:
  • The prime factors are: 2 x 17 x 31 x 67 x 439
    • or also written as { 2, 17, 31, 67, 439 }
  • Written in exponential form: 21 x 171 x 311 x 671 x 4391

Why is the prime factorization of 31,001,302 written as 21 x 171 x 311 x 671 x 4391?

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 31,001,302

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 31,001,302 by 2

31,001,302 ÷ 2 = 15,500,651 - No remainder! 2 is one of the factors!
15,500,651 ÷ 2 = 7,750,325.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
15,500,651 ÷ 3 = 5,166,883.6667 - This has a remainder. 3 is not a factor.
15,500,651 ÷ 5 = 3,100,130.2 - This has a remainder. 5 is not a factor.
15,500,651 ÷ 7 = 2,214,378.7143 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
15,500,651 ÷ 17 = 911,803 - No remainder! 17 is one of the factors!
911,803 ÷ 17 = 53,635.4706 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
911,803 ÷ 19 = 47,989.6316 - This has a remainder. 19 is not a factor.
911,803 ÷ 23 = 39,643.6087 - This has a remainder. 23 is not a factor.
911,803 ÷ 29 = 31,441.4828 - This has a remainder. 29 is not a factor.
911,803 ÷ 31 = 29,413 - No remainder! 31 is one of the factors!
29,413 ÷ 31 = 948.8065 - There is a remainder. We can't divide by 31 evenly anymore. Let's try the next prime number
29,413 ÷ 37 = 794.9459 - This has a remainder. 37 is not a factor.
29,413 ÷ 41 = 717.3902 - This has a remainder. 41 is not a factor.
29,413 ÷ 43 = 684.0233 - This has a remainder. 43 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
29,413 ÷ 67 = 439 - No remainder! 67 is one of the factors!
439 ÷ 67 = 6.5522 - There is a remainder. We can't divide by 67 evenly anymore. Let's try the next prime number
439 ÷ 71 = 6.1831 - This has a remainder. 71 is not a factor.
439 ÷ 73 = 6.0137 - This has a remainder. 73 is not a factor.
439 ÷ 79 = 5.557 - This has a remainder. 79 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
439 ÷ 439 = 1 - No remainder! 439 is one of the factors!

The orange divisor(s) above are the prime factors of the number 31,001,302. If we put all of it together we have the factors 2 x 17 x 31 x 67 x 439 = 31,001,302. It can also be written in exponential form as 21 x 171 x 311 x 671 x 4391.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 31,001,302.

31,001,302
Factor Arrows
215,500,651
Factor Arrows
17911,803
Factor Arrows
3129,413
Factor Arrows
67439

More Prime Factorization Examples

31,001,30031,001,30131,001,30331,001,304
22 x 52 x 111 x 28,183132 x 371 x 93,09715711 x 54,293123 x 31 x 3371 x 3,8331

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