Q: What is the prime factorization of the number 338,346,600?

 A:
  • The prime factors are: 2 x 2 x 2 x 3 x 5 x 5 x 643 x 877
    • or also written as { 2, 2, 2, 3, 5, 5, 643, 877 }
  • Written in exponential form: 23 x 31 x 52 x 6431 x 8771

Why is the prime factorization of 338,346,600 written as 23 x 31 x 52 x 6431 x 8771?

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 338,346,600

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 338,346,600 by 2

338,346,600 ÷ 2 = 169,173,300 - No remainder! 2 is one of the factors!
169,173,300 ÷ 2 = 84,586,650 - No remainder! 2 is one of the factors!
84,586,650 ÷ 2 = 42,293,325 - No remainder! 2 is one of the factors!
42,293,325 ÷ 2 = 21,146,662.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
42,293,325 ÷ 3 = 14,097,775 - No remainder! 3 is one of the factors!
14,097,775 ÷ 3 = 4,699,258.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
14,097,775 ÷ 5 = 2,819,555 - No remainder! 5 is one of the factors!
2,819,555 ÷ 5 = 563,911 - No remainder! 5 is one of the factors!
563,911 ÷ 5 = 112,782.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
563,911 ÷ 7 = 80,558.7143 - This has a remainder. 7 is not a factor.
563,911 ÷ 11 = 51,264.6364 - This has a remainder. 11 is not a factor.
563,911 ÷ 13 = 43,377.7692 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
563,911 ÷ 643 = 877 - No remainder! 643 is one of the factors!
877 ÷ 643 = 1.3639 - There is a remainder. We can't divide by 643 evenly anymore. Let's try the next prime number
877 ÷ 647 = 1.3555 - This has a remainder. 647 is not a factor.
877 ÷ 653 = 1.343 - This has a remainder. 653 is not a factor.
877 ÷ 659 = 1.3308 - This has a remainder. 659 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
877 ÷ 877 = 1 - No remainder! 877 is one of the factors!

The orange divisor(s) above are the prime factors of the number 338,346,600. If we put all of it together we have the factors 2 x 2 x 2 x 3 x 5 x 5 x 643 x 877 = 338,346,600. It can also be written in exponential form as 23 x 31 x 52 x 6431 x 8771.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 338,346,600.

338,346,600
Factor Arrows
2169,173,300
Factor Arrows
284,586,650
Factor Arrows
242,293,325
Factor Arrows
314,097,775
Factor Arrows
52,819,555
Factor Arrows
5563,911
Factor Arrows
643877

More Prime Factorization Examples

338,346,598338,346,599338,346,601338,346,602
21 x 1631 x 1,037,873111,9091 x 28,411117,0771 x 19,813121 x 111 x 15,379,3911

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