Q: What is the prime factorization of the number 287,214,564?

 A:
  • The prime factors are: 2 x 2 x 3 x 7 x 13 x 19 x 109 x 127
    • or also written as { 2, 2, 3, 7, 13, 19, 109, 127 }
  • Written in exponential form: 22 x 31 x 71 x 131 x 191 x 1091 x 1271

Why is the prime factorization of 287,214,564 written as 22 x 31 x 71 x 131 x 191 x 1091 x 1271?

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 287,214,564

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 287,214,564 by 2

287,214,564 ÷ 2 = 143,607,282 - No remainder! 2 is one of the factors!
143,607,282 ÷ 2 = 71,803,641 - No remainder! 2 is one of the factors!
71,803,641 ÷ 2 = 35,901,820.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
71,803,641 ÷ 3 = 23,934,547 - No remainder! 3 is one of the factors!
23,934,547 ÷ 3 = 7,978,182.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
23,934,547 ÷ 5 = 4,786,909.4 - This has a remainder. 5 is not a factor.
23,934,547 ÷ 7 = 3,419,221 - No remainder! 7 is one of the factors!
3,419,221 ÷ 7 = 488,460.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
3,419,221 ÷ 11 = 310,838.2727 - This has a remainder. 11 is not a factor.
3,419,221 ÷ 13 = 263,017 - No remainder! 13 is one of the factors!
263,017 ÷ 13 = 20,232.0769 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
263,017 ÷ 17 = 15,471.5882 - This has a remainder. 17 is not a factor.
263,017 ÷ 19 = 13,843 - No remainder! 19 is one of the factors!
13,843 ÷ 19 = 728.5789 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
13,843 ÷ 23 = 601.8696 - This has a remainder. 23 is not a factor.
13,843 ÷ 29 = 477.3448 - This has a remainder. 29 is not a factor.
13,843 ÷ 31 = 446.5484 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
13,843 ÷ 109 = 127 - No remainder! 109 is one of the factors!
127 ÷ 109 = 1.1651 - There is a remainder. We can't divide by 109 evenly anymore. Let's try the next prime number
127 ÷ 113 = 1.1239 - This has a remainder. 113 is not a factor.
127 ÷ 127 = 1 - No remainder! 127 is one of the factors!

The orange divisor(s) above are the prime factors of the number 287,214,564. If we put all of it together we have the factors 2 x 2 x 3 x 7 x 13 x 19 x 109 x 127 = 287,214,564. It can also be written in exponential form as 22 x 31 x 71 x 131 x 191 x 1091 x 1271.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 287,214,564.

287,214,564
Factor Arrows
2143,607,282
Factor Arrows
271,803,641
Factor Arrows
323,934,547
Factor Arrows
73,419,221
Factor Arrows
13263,017
Factor Arrows
1913,843
Factor Arrows
109127

More Prime Factorization Examples

287,214,562287,214,563287,214,565287,214,566
21 x 143,607,2811287,214,563151 x 111 x 5,222,083121 x 311 x 2,0991 x 2,2071

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