Q: What is the prime factorization of the number 313,220,138?

 A:
  • The prime factors are: 2 x 7 x 11 x 17 x 181 x 661
    • or also written as { 2, 7, 11, 17, 181, 661 }
  • Written in exponential form: 21 x 71 x 111 x 171 x 1811 x 6611

Why is the prime factorization of 313,220,138 written as 21 x 71 x 111 x 171 x 1811 x 6611?

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 313,220,138

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 313,220,138 by 2

313,220,138 ÷ 2 = 156,610,069 - No remainder! 2 is one of the factors!
156,610,069 ÷ 2 = 78,305,034.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
156,610,069 ÷ 3 = 52,203,356.3333 - This has a remainder. 3 is not a factor.
156,610,069 ÷ 5 = 31,322,013.8 - This has a remainder. 5 is not a factor.
156,610,069 ÷ 7 = 22,372,867 - No remainder! 7 is one of the factors!
22,372,867 ÷ 7 = 3,196,123.8571 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
22,372,867 ÷ 11 = 2,033,897 - No remainder! 11 is one of the factors!
2,033,897 ÷ 11 = 184,899.7273 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
2,033,897 ÷ 13 = 156,453.6154 - This has a remainder. 13 is not a factor.
2,033,897 ÷ 17 = 119,641 - No remainder! 17 is one of the factors!
119,641 ÷ 17 = 7,037.7059 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
119,641 ÷ 19 = 6,296.8947 - This has a remainder. 19 is not a factor.
119,641 ÷ 23 = 5,201.7826 - This has a remainder. 23 is not a factor.
119,641 ÷ 29 = 4,125.5517 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
119,641 ÷ 181 = 661 - No remainder! 181 is one of the factors!
661 ÷ 181 = 3.6519 - There is a remainder. We can't divide by 181 evenly anymore. Let's try the next prime number
661 ÷ 191 = 3.4607 - This has a remainder. 191 is not a factor.
661 ÷ 193 = 3.4249 - This has a remainder. 193 is not a factor.
661 ÷ 197 = 3.3553 - This has a remainder. 197 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
661 ÷ 661 = 1 - No remainder! 661 is one of the factors!

The orange divisor(s) above are the prime factors of the number 313,220,138. If we put all of it together we have the factors 2 x 7 x 11 x 17 x 181 x 661 = 313,220,138. It can also be written in exponential form as 21 x 71 x 111 x 171 x 1811 x 6611.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 313,220,138.

313,220,138
Factor Arrows
2156,610,069
Factor Arrows
722,372,867
Factor Arrows
112,033,897
Factor Arrows
17119,641
Factor Arrows
181661

More Prime Factorization Examples

313,220,136313,220,137313,220,139313,220,140
23 x 31 x 1911 x 68,32911071 x 1,0871 x 2,693131 x 2511 x 415,963122 x 51 x 15,661,0071

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