Q: What is the prime factorization of the number 210,120,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 x 5 x 17 x 103
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 5, 5, 5, 5, 17, 103 }
  • Written in exponential form: 26 x 31 x 54 x 171 x 1031

Why is the prime factorization of 210,120,000 written as 26 x 31 x 54 x 171 x 1031?

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 210,120,000

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 210,120,000 by 2

210,120,000 ÷ 2 = 105,060,000 - No remainder! 2 is one of the factors!
105,060,000 ÷ 2 = 52,530,000 - No remainder! 2 is one of the factors!
52,530,000 ÷ 2 = 26,265,000 - No remainder! 2 is one of the factors!
26,265,000 ÷ 2 = 13,132,500 - No remainder! 2 is one of the factors!
13,132,500 ÷ 2 = 6,566,250 - No remainder! 2 is one of the factors!
6,566,250 ÷ 2 = 3,283,125 - No remainder! 2 is one of the factors!
3,283,125 ÷ 2 = 1,641,562.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
3,283,125 ÷ 3 = 1,094,375 - No remainder! 3 is one of the factors!
1,094,375 ÷ 3 = 364,791.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
1,094,375 ÷ 5 = 218,875 - No remainder! 5 is one of the factors!
218,875 ÷ 5 = 43,775 - No remainder! 5 is one of the factors!
43,775 ÷ 5 = 8,755 - No remainder! 5 is one of the factors!
8,755 ÷ 5 = 1,751 - No remainder! 5 is one of the factors!
1,751 ÷ 5 = 350.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
1,751 ÷ 7 = 250.1429 - This has a remainder. 7 is not a factor.
1,751 ÷ 11 = 159.1818 - This has a remainder. 11 is not a factor.
1,751 ÷ 13 = 134.6923 - This has a remainder. 13 is not a factor.
1,751 ÷ 17 = 103 - No remainder! 17 is one of the factors!
103 ÷ 17 = 6.0588 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
103 ÷ 19 = 5.4211 - This has a remainder. 19 is not a factor.
103 ÷ 23 = 4.4783 - This has a remainder. 23 is not a factor.
103 ÷ 29 = 3.5517 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
103 ÷ 103 = 1 - No remainder! 103 is one of the factors!

The orange divisor(s) above are the prime factors of the number 210,120,000. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 x 5 x 17 x 103 = 210,120,000. It can also be written in exponential form as 26 x 31 x 54 x 171 x 1031.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 210,120,000.

210,120,000
Factor Arrows
2105,060,000
Factor Arrows
252,530,000
Factor Arrows
226,265,000
Factor Arrows
213,132,500
Factor Arrows
26,566,250
Factor Arrows
23,283,125
Factor Arrows
31,094,375
Factor Arrows
5218,875
Factor Arrows
543,775
Factor Arrows
58,755
Factor Arrows
51,751
Factor Arrows
17103

More Prime Factorization Examples

210,119,998210,119,999210,120,001210,120,002
21 x 111 x 411 x 2811 x 82918391 x 250,441171 x 131 x 2,309,011121 x 6611 x 158,9411

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