Q: What is the prime factorization of the number 242,515,500?

 A:
  • The prime factors are: 2 x 2 x 3 x 5 x 5 x 5 x 107 x 1,511
    • or also written as { 2, 2, 3, 5, 5, 5, 107, 1,511 }
  • Written in exponential form: 22 x 31 x 53 x 1071 x 1,5111

Why is the prime factorization of 242,515,500 written as 22 x 31 x 53 x 1071 x 1,5111?

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 242,515,500

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 242,515,500 by 2

242,515,500 ÷ 2 = 121,257,750 - No remainder! 2 is one of the factors!
121,257,750 ÷ 2 = 60,628,875 - No remainder! 2 is one of the factors!
60,628,875 ÷ 2 = 30,314,437.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
60,628,875 ÷ 3 = 20,209,625 - No remainder! 3 is one of the factors!
20,209,625 ÷ 3 = 6,736,541.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
20,209,625 ÷ 5 = 4,041,925 - No remainder! 5 is one of the factors!
4,041,925 ÷ 5 = 808,385 - No remainder! 5 is one of the factors!
808,385 ÷ 5 = 161,677 - No remainder! 5 is one of the factors!
161,677 ÷ 5 = 32,335.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
161,677 ÷ 7 = 23,096.7143 - This has a remainder. 7 is not a factor.
161,677 ÷ 11 = 14,697.9091 - This has a remainder. 11 is not a factor.
161,677 ÷ 13 = 12,436.6923 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
161,677 ÷ 107 = 1,511 - No remainder! 107 is one of the factors!
1,511 ÷ 107 = 14.1215 - There is a remainder. We can't divide by 107 evenly anymore. Let's try the next prime number
1,511 ÷ 109 = 13.8624 - This has a remainder. 109 is not a factor.
1,511 ÷ 113 = 13.3717 - This has a remainder. 113 is not a factor.
1,511 ÷ 127 = 11.8976 - This has a remainder. 127 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,511 ÷ 1,511 = 1 - No remainder! 1,511 is one of the factors!

The orange divisor(s) above are the prime factors of the number 242,515,500. If we put all of it together we have the factors 2 x 2 x 3 x 5 x 5 x 5 x 107 x 1,511 = 242,515,500. It can also be written in exponential form as 22 x 31 x 53 x 1071 x 1,5111.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 242,515,500.

242,515,500
Factor Arrows
2121,257,750
Factor Arrows
260,628,875
Factor Arrows
320,209,625
Factor Arrows
54,041,925
Factor Arrows
5808,385
Factor Arrows
5161,677
Factor Arrows
1071,511

More Prime Factorization Examples

242,515,498242,515,499242,515,501242,515,502
21 x 5,7911 x 20,9391242,515,4991371 x 1931 x 33,961121 x 121,257,7511

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