Q: What is the prime factorization of the number 214,376,754?

 A:
  • The prime factors are: 2 x 3 x 1,307 x 27,337
    • or also written as { 2, 3, 1,307, 27,337 }
  • Written in exponential form: 21 x 31 x 1,3071 x 27,3371

Why is the prime factorization of 214,376,754 written as 21 x 31 x 1,3071 x 27,3371?

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 214,376,754

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 214,376,754 by 2

214,376,754 ÷ 2 = 107,188,377 - No remainder! 2 is one of the factors!
107,188,377 ÷ 2 = 53,594,188.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
107,188,377 ÷ 3 = 35,729,459 - No remainder! 3 is one of the factors!
35,729,459 ÷ 3 = 11,909,819.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
35,729,459 ÷ 5 = 7,145,891.8 - This has a remainder. 5 is not a factor.
35,729,459 ÷ 7 = 5,104,208.4286 - This has a remainder. 7 is not a factor.
35,729,459 ÷ 11 = 3,248,132.6364 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
35,729,459 ÷ 1,307 = 27,337 - No remainder! 1,307 is one of the factors!
27,337 ÷ 1,307 = 20.9158 - There is a remainder. We can't divide by 1307 evenly anymore. Let's try the next prime number
27,337 ÷ 1,319 = 20.7255 - This has a remainder. 1,319 is not a factor.
27,337 ÷ 1,321 = 20.6942 - This has a remainder. 1,321 is not a factor.
27,337 ÷ 1,327 = 20.6006 - This has a remainder. 1,327 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
27,337 ÷ 27,337 = 1 - No remainder! 27,337 is one of the factors!

The orange divisor(s) above are the prime factors of the number 214,376,754. If we put all of it together we have the factors 2 x 3 x 1,307 x 27,337 = 214,376,754. It can also be written in exponential form as 21 x 31 x 1,3071 x 27,3371.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 214,376,754.

214,376,754
Factor Arrows
2107,188,377
Factor Arrows
335,729,459
Factor Arrows
1,30727,337

More Prime Factorization Examples

214,376,752214,376,753214,376,755214,376,756
24 x 13,398,5471191 x 611 x 184,967151 x 42,875,351122 x 111 x 1731 x 28,1631

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