Q: What is the prime factorization of the number 323,000,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5 x 5 x 5 x 17 x 19
    • or also written as { 2, 2, 2, 2, 2, 2, 5, 5, 5, 5, 5, 5, 17, 19 }
  • Written in exponential form: 26 x 56 x 171 x 191

Why is the prime factorization of 323,000,000 written as 26 x 56 x 171 x 191?

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 323,000,000

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 323,000,000 by 2

323,000,000 ÷ 2 = 161,500,000 - No remainder! 2 is one of the factors!
161,500,000 ÷ 2 = 80,750,000 - No remainder! 2 is one of the factors!
80,750,000 ÷ 2 = 40,375,000 - No remainder! 2 is one of the factors!
40,375,000 ÷ 2 = 20,187,500 - No remainder! 2 is one of the factors!
20,187,500 ÷ 2 = 10,093,750 - No remainder! 2 is one of the factors!
10,093,750 ÷ 2 = 5,046,875 - No remainder! 2 is one of the factors!
5,046,875 ÷ 2 = 2,523,437.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
5,046,875 ÷ 3 = 1,682,291.6667 - This has a remainder. 3 is not a factor.
5,046,875 ÷ 5 = 1,009,375 - No remainder! 5 is one of the factors!
1,009,375 ÷ 5 = 201,875 - No remainder! 5 is one of the factors!
201,875 ÷ 5 = 40,375 - No remainder! 5 is one of the factors!
40,375 ÷ 5 = 8,075 - No remainder! 5 is one of the factors!
8,075 ÷ 5 = 1,615 - No remainder! 5 is one of the factors!
1,615 ÷ 5 = 323 - No remainder! 5 is one of the factors!
323 ÷ 5 = 64.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
323 ÷ 7 = 46.1429 - This has a remainder. 7 is not a factor.
323 ÷ 11 = 29.3636 - This has a remainder. 11 is not a factor.
323 ÷ 13 = 24.8462 - This has a remainder. 13 is not a factor.
323 ÷ 17 = 19 - No remainder! 17 is one of the factors!
19 ÷ 17 = 1.1176 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
19 ÷ 19 = 1 - No remainder! 19 is one of the factors!

The orange divisor(s) above are the prime factors of the number 323,000,000. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5 x 5 x 5 x 17 x 19 = 323,000,000. It can also be written in exponential form as 26 x 56 x 171 x 191.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 323,000,000.

323,000,000
Factor Arrows
2161,500,000
Factor Arrows
280,750,000
Factor Arrows
240,375,000
Factor Arrows
220,187,500
Factor Arrows
210,093,750
Factor Arrows
25,046,875
Factor Arrows
51,009,375
Factor Arrows
5201,875
Factor Arrows
540,375
Factor Arrows
58,075
Factor Arrows
51,615
Factor Arrows
5323
Factor Arrows
1719

More Prime Factorization Examples

322,999,998322,999,999323,000,001323,000,002
21 x 31 x 53,833,333171 x 291 x 1391 x 11,447133 x 11,962,963121 x 131 x 611 x 203,6571

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