Q: What is the prime factorization of the number 242,242,560?

 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 3 x 5 x 7 x 751
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 5, 7, 751 }
  • Written in exponential form: 210 x 32 x 51 x 71 x 7511

Why is the prime factorization of 242,242,560 written as 210 x 32 x 51 x 71 x 7511?

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 242,242,560

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 242,242,560 by 2

242,242,560 ÷ 2 = 121,121,280 - No remainder! 2 is one of the factors!
121,121,280 ÷ 2 = 60,560,640 - No remainder! 2 is one of the factors!
60,560,640 ÷ 2 = 30,280,320 - No remainder! 2 is one of the factors!
30,280,320 ÷ 2 = 15,140,160 - No remainder! 2 is one of the factors!
15,140,160 ÷ 2 = 7,570,080 - No remainder! 2 is one of the factors!
7,570,080 ÷ 2 = 3,785,040 - No remainder! 2 is one of the factors!
3,785,040 ÷ 2 = 1,892,520 - No remainder! 2 is one of the factors!
1,892,520 ÷ 2 = 946,260 - No remainder! 2 is one of the factors!
946,260 ÷ 2 = 473,130 - No remainder! 2 is one of the factors!
473,130 ÷ 2 = 236,565 - No remainder! 2 is one of the factors!
236,565 ÷ 2 = 118,282.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
236,565 ÷ 3 = 78,855 - No remainder! 3 is one of the factors!
78,855 ÷ 3 = 26,285 - No remainder! 3 is one of the factors!
26,285 ÷ 3 = 8,761.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
26,285 ÷ 5 = 5,257 - No remainder! 5 is one of the factors!
5,257 ÷ 5 = 1,051.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
5,257 ÷ 7 = 751 - No remainder! 7 is one of the factors!
751 ÷ 7 = 107.2857 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
751 ÷ 11 = 68.2727 - This has a remainder. 11 is not a factor.
751 ÷ 13 = 57.7692 - This has a remainder. 13 is not a factor.
751 ÷ 17 = 44.1765 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
751 ÷ 751 = 1 - No remainder! 751 is one of the factors!

The orange divisor(s) above are the prime factors of the number 242,242,560. 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 3 x 5 x 7 x 751 = 242,242,560. It can also be written in exponential form as 210 x 32 x 51 x 71 x 7511.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 242,242,560.

242,242,560
Factor Arrows
2121,121,280
Factor Arrows
260,560,640
Factor Arrows
230,280,320
Factor Arrows
215,140,160
Factor Arrows
27,570,080
Factor Arrows
23,785,040
Factor Arrows
21,892,520
Factor Arrows
2946,260
Factor Arrows
2473,130
Factor Arrows
2236,565
Factor Arrows
378,855
Factor Arrows
326,285
Factor Arrows
55,257
Factor Arrows
7751

More Prime Factorization Examples

242,242,558242,242,559242,242,561242,242,562
21 x 1731 x 4211 x 1,6631131 x 471 x 2111 x 1,8791111 x 1491 x 147,799121 x 971 x 1,248,6731

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