Q: What is the prime factorization of the number 32,860,260?

 A:
  • The prime factors are: 2 x 2 x 3 x 3 x 5 x 311 x 587
    • or also written as { 2, 2, 3, 3, 5, 311, 587 }
  • Written in exponential form: 22 x 32 x 51 x 3111 x 5871

Why is the prime factorization of 32,860,260 written as 22 x 32 x 51 x 3111 x 5871?

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 32,860,260

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 32,860,260 by 2

32,860,260 ÷ 2 = 16,430,130 - No remainder! 2 is one of the factors!
16,430,130 ÷ 2 = 8,215,065 - No remainder! 2 is one of the factors!
8,215,065 ÷ 2 = 4,107,532.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
8,215,065 ÷ 3 = 2,738,355 - No remainder! 3 is one of the factors!
2,738,355 ÷ 3 = 912,785 - No remainder! 3 is one of the factors!
912,785 ÷ 3 = 304,261.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
912,785 ÷ 5 = 182,557 - No remainder! 5 is one of the factors!
182,557 ÷ 5 = 36,511.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
182,557 ÷ 7 = 26,079.5714 - This has a remainder. 7 is not a factor.
182,557 ÷ 11 = 16,596.0909 - This has a remainder. 11 is not a factor.
182,557 ÷ 13 = 14,042.8462 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
182,557 ÷ 311 = 587 - No remainder! 311 is one of the factors!
587 ÷ 311 = 1.8875 - There is a remainder. We can't divide by 311 evenly anymore. Let's try the next prime number
587 ÷ 313 = 1.8754 - This has a remainder. 313 is not a factor.
587 ÷ 317 = 1.8517 - This has a remainder. 317 is not a factor.
587 ÷ 331 = 1.7734 - This has a remainder. 331 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
587 ÷ 587 = 1 - No remainder! 587 is one of the factors!

The orange divisor(s) above are the prime factors of the number 32,860,260. If we put all of it together we have the factors 2 x 2 x 3 x 3 x 5 x 311 x 587 = 32,860,260. It can also be written in exponential form as 22 x 32 x 51 x 3111 x 5871.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 32,860,260.

32,860,260
Factor Arrows
216,430,130
Factor Arrows
28,215,065
Factor Arrows
32,738,355
Factor Arrows
3912,785
Factor Arrows
5182,557
Factor Arrows
311587

More Prime Factorization Examples

32,860,25832,860,25932,860,26132,860,262
21 x 16,430,12914,1591 x 7,901171 x 231 x 204,101121 x 2831 x 58,0571

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