Q: What is the prime factorization of the number 200,687,585?

 A:
  • The prime factors are: 5 x 7 x 7 x 7 x 7 x 73 x 229
    • or also written as { 5, 7, 7, 7, 7, 73, 229 }
  • Written in exponential form: 51 x 74 x 731 x 2291

Why is the prime factorization of 200,687,585 written as 51 x 74 x 731 x 2291?

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 200,687,585

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 200,687,585 by 2

200,687,585 ÷ 2 = 100,343,792.5 - This has a remainder. Let's try another prime number.
200,687,585 ÷ 3 = 66,895,861.6667 - This has a remainder. Let's try another prime number.
200,687,585 ÷ 5 = 40,137,517 - No remainder! 5 is one of the factors!
40,137,517 ÷ 5 = 8,027,503.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
40,137,517 ÷ 7 = 5,733,931 - No remainder! 7 is one of the factors!
5,733,931 ÷ 7 = 819,133 - No remainder! 7 is one of the factors!
819,133 ÷ 7 = 117,019 - No remainder! 7 is one of the factors!
117,019 ÷ 7 = 16,717 - No remainder! 7 is one of the factors!
16,717 ÷ 7 = 2,388.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
16,717 ÷ 11 = 1,519.7273 - This has a remainder. 11 is not a factor.
16,717 ÷ 13 = 1,285.9231 - This has a remainder. 13 is not a factor.
16,717 ÷ 17 = 983.3529 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
16,717 ÷ 73 = 229 - No remainder! 73 is one of the factors!
229 ÷ 73 = 3.137 - There is a remainder. We can't divide by 73 evenly anymore. Let's try the next prime number
229 ÷ 79 = 2.8987 - This has a remainder. 79 is not a factor.
229 ÷ 83 = 2.759 - This has a remainder. 83 is not a factor.
229 ÷ 89 = 2.573 - This has a remainder. 89 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
229 ÷ 229 = 1 - No remainder! 229 is one of the factors!

The orange divisor(s) above are the prime factors of the number 200,687,585. If we put all of it together we have the factors 5 x 7 x 7 x 7 x 7 x 73 x 229 = 200,687,585. It can also be written in exponential form as 51 x 74 x 731 x 2291.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 200,687,585.

200,687,585
Factor Arrows
540,137,517
Factor Arrows
75,733,931
Factor Arrows
7819,133
Factor Arrows
7117,019
Factor Arrows
716,717
Factor Arrows
73229

More Prime Factorization Examples

200,687,583200,687,584200,687,586200,687,587
31 x 311 x 2391 x 9,029125 x 171 x 368,911121 x 31 x 111 x 2111 x 14,4111200,687,5871

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