Q: What is the prime factorization of the number 10,122,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 x 7 x 241
    • or also written as { 2, 2, 2, 2, 3, 5, 5, 5, 7, 241 }
  • Written in exponential form: 24 x 31 x 53 x 71 x 2411

Why is the prime factorization of 10,122,000 written as 24 x 31 x 53 x 71 x 2411?

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 10,122,000

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 10,122,000 by 2

10,122,000 ÷ 2 = 5,061,000 - No remainder! 2 is one of the factors!
5,061,000 ÷ 2 = 2,530,500 - No remainder! 2 is one of the factors!
2,530,500 ÷ 2 = 1,265,250 - No remainder! 2 is one of the factors!
1,265,250 ÷ 2 = 632,625 - No remainder! 2 is one of the factors!
632,625 ÷ 2 = 316,312.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
632,625 ÷ 3 = 210,875 - No remainder! 3 is one of the factors!
210,875 ÷ 3 = 70,291.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
210,875 ÷ 5 = 42,175 - No remainder! 5 is one of the factors!
42,175 ÷ 5 = 8,435 - No remainder! 5 is one of the factors!
8,435 ÷ 5 = 1,687 - No remainder! 5 is one of the factors!
1,687 ÷ 5 = 337.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
1,687 ÷ 7 = 241 - No remainder! 7 is one of the factors!
241 ÷ 7 = 34.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
241 ÷ 11 = 21.9091 - This has a remainder. 11 is not a factor.
241 ÷ 13 = 18.5385 - This has a remainder. 13 is not a factor.
241 ÷ 17 = 14.1765 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
241 ÷ 241 = 1 - No remainder! 241 is one of the factors!

The orange divisor(s) above are the prime factors of the number 10,122,000. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 x 7 x 241 = 10,122,000. It can also be written in exponential form as 24 x 31 x 53 x 71 x 2411.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 10,122,000.

10,122,000
Factor Arrows
25,061,000
Factor Arrows
22,530,500
Factor Arrows
21,265,250
Factor Arrows
2632,625
Factor Arrows
3210,875
Factor Arrows
542,175
Factor Arrows
58,435
Factor Arrows
51,687
Factor Arrows
7241

More Prime Factorization Examples

10,121,99810,121,99910,122,00110,122,002
21 x 411 x 123,439110,121,9991231 x 440,087121 x 111 x 460,0911

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