Q: What is the prime factorization of the number 31,818,240?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 x 1,381
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 5, 1,381 }
  • Written in exponential form: 29 x 32 x 51 x 1,3811

Why is the prime factorization of 31,818,240 written as 29 x 32 x 51 x 1,3811?

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 31,818,240

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 31,818,240 by 2

31,818,240 ÷ 2 = 15,909,120 - No remainder! 2 is one of the factors!
15,909,120 ÷ 2 = 7,954,560 - No remainder! 2 is one of the factors!
7,954,560 ÷ 2 = 3,977,280 - No remainder! 2 is one of the factors!
3,977,280 ÷ 2 = 1,988,640 - No remainder! 2 is one of the factors!
1,988,640 ÷ 2 = 994,320 - No remainder! 2 is one of the factors!
994,320 ÷ 2 = 497,160 - No remainder! 2 is one of the factors!
497,160 ÷ 2 = 248,580 - No remainder! 2 is one of the factors!
248,580 ÷ 2 = 124,290 - No remainder! 2 is one of the factors!
124,290 ÷ 2 = 62,145 - No remainder! 2 is one of the factors!
62,145 ÷ 2 = 31,072.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
62,145 ÷ 3 = 20,715 - No remainder! 3 is one of the factors!
20,715 ÷ 3 = 6,905 - No remainder! 3 is one of the factors!
6,905 ÷ 3 = 2,301.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
6,905 ÷ 5 = 1,381 - No remainder! 5 is one of the factors!
1,381 ÷ 5 = 276.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
1,381 ÷ 7 = 197.2857 - This has a remainder. 7 is not a factor.
1,381 ÷ 11 = 125.5455 - This has a remainder. 11 is not a factor.
1,381 ÷ 13 = 106.2308 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,381 ÷ 1,381 = 1 - No remainder! 1,381 is one of the factors!

The orange divisor(s) above are the prime factors of the number 31,818,240. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 x 1,381 = 31,818,240. It can also be written in exponential form as 29 x 32 x 51 x 1,3811.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 31,818,240.

31,818,240
Factor Arrows
215,909,120
Factor Arrows
27,954,560
Factor Arrows
23,977,280
Factor Arrows
21,988,640
Factor Arrows
2994,320
Factor Arrows
2497,160
Factor Arrows
2248,580
Factor Arrows
2124,290
Factor Arrows
262,145
Factor Arrows
320,715
Factor Arrows
36,905
Factor Arrows
51,381

More Prime Factorization Examples

31,818,23831,818,23931,818,24131,818,242
21 x 6831 x 23,293131,818,239171 x 131 x 3171 x 1,103121 x 1,6931 x 9,3971

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