Q: What is the prime factorization of the number 670,467,072?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 11 x 19,841
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 11, 19,841 }
  • Written in exponential form: 210 x 31 x 111 x 19,8411

Why is the prime factorization of 670,467,072 written as 210 x 31 x 111 x 19,8411?

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 670,467,072

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 670,467,072 by 2

670,467,072 ÷ 2 = 335,233,536 - No remainder! 2 is one of the factors!
335,233,536 ÷ 2 = 167,616,768 - No remainder! 2 is one of the factors!
167,616,768 ÷ 2 = 83,808,384 - No remainder! 2 is one of the factors!
83,808,384 ÷ 2 = 41,904,192 - No remainder! 2 is one of the factors!
41,904,192 ÷ 2 = 20,952,096 - No remainder! 2 is one of the factors!
20,952,096 ÷ 2 = 10,476,048 - No remainder! 2 is one of the factors!
10,476,048 ÷ 2 = 5,238,024 - No remainder! 2 is one of the factors!
5,238,024 ÷ 2 = 2,619,012 - No remainder! 2 is one of the factors!
2,619,012 ÷ 2 = 1,309,506 - No remainder! 2 is one of the factors!
1,309,506 ÷ 2 = 654,753 - No remainder! 2 is one of the factors!
654,753 ÷ 2 = 327,376.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
654,753 ÷ 3 = 218,251 - No remainder! 3 is one of the factors!
218,251 ÷ 3 = 72,750.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
218,251 ÷ 5 = 43,650.2 - This has a remainder. 5 is not a factor.
218,251 ÷ 7 = 31,178.7143 - This has a remainder. 7 is not a factor.
218,251 ÷ 11 = 19,841 - No remainder! 11 is one of the factors!
19,841 ÷ 11 = 1,803.7273 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
19,841 ÷ 13 = 1,526.2308 - This has a remainder. 13 is not a factor.
19,841 ÷ 17 = 1,167.1176 - This has a remainder. 17 is not a factor.
19,841 ÷ 19 = 1,044.2632 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
19,841 ÷ 19,841 = 1 - No remainder! 19,841 is one of the factors!

The orange divisor(s) above are the prime factors of the number 670,467,072. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 11 x 19,841 = 670,467,072. It can also be written in exponential form as 210 x 31 x 111 x 19,8411.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 670,467,072.

670,467,072
Factor Arrows
2335,233,536
Factor Arrows
2167,616,768
Factor Arrows
283,808,384
Factor Arrows
241,904,192
Factor Arrows
220,952,096
Factor Arrows
210,476,048
Factor Arrows
25,238,024
Factor Arrows
22,619,012
Factor Arrows
21,309,506
Factor Arrows
2654,753
Factor Arrows
3218,251
Factor Arrows
1119,841

More Prime Factorization Examples

670,467,070670,467,071670,467,073670,467,074
21 x 51 x 71 x 131 x 311 x 23,76711491 x 4,499,77918,4431 x 79,411121 x 335,233,5371

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