Q: What is the prime factorization of the number 111,255,552?

 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 2 x 2 x 2 x 3 x 3 x 3 x 503
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 503 }
  • Written in exponential form: 213 x 33 x 5031

Why is the prime factorization of 111,255,552 written as 213 x 33 x 5031?

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 111,255,552

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 111,255,552 by 2

111,255,552 ÷ 2 = 55,627,776 - No remainder! 2 is one of the factors!
55,627,776 ÷ 2 = 27,813,888 - No remainder! 2 is one of the factors!
27,813,888 ÷ 2 = 13,906,944 - No remainder! 2 is one of the factors!
13,906,944 ÷ 2 = 6,953,472 - No remainder! 2 is one of the factors!
6,953,472 ÷ 2 = 3,476,736 - No remainder! 2 is one of the factors!
3,476,736 ÷ 2 = 1,738,368 - No remainder! 2 is one of the factors!
1,738,368 ÷ 2 = 869,184 - No remainder! 2 is one of the factors!
869,184 ÷ 2 = 434,592 - No remainder! 2 is one of the factors!
434,592 ÷ 2 = 217,296 - No remainder! 2 is one of the factors!
217,296 ÷ 2 = 108,648 - No remainder! 2 is one of the factors!
108,648 ÷ 2 = 54,324 - No remainder! 2 is one of the factors!
54,324 ÷ 2 = 27,162 - No remainder! 2 is one of the factors!
27,162 ÷ 2 = 13,581 - No remainder! 2 is one of the factors!
13,581 ÷ 2 = 6,790.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
13,581 ÷ 3 = 4,527 - No remainder! 3 is one of the factors!
4,527 ÷ 3 = 1,509 - No remainder! 3 is one of the factors!
1,509 ÷ 3 = 503 - No remainder! 3 is one of the factors!
503 ÷ 3 = 167.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
503 ÷ 5 = 100.6 - This has a remainder. 5 is not a factor.
503 ÷ 7 = 71.8571 - This has a remainder. 7 is not a factor.
503 ÷ 11 = 45.7273 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
503 ÷ 503 = 1 - No remainder! 503 is one of the factors!

The orange divisor(s) above are the prime factors of the number 111,255,552. 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 2 x 2 x 2 x 3 x 3 x 3 x 503 = 111,255,552. It can also be written in exponential form as 213 x 33 x 5031.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 111,255,552.

111,255,552
Factor Arrows
255,627,776
Factor Arrows
227,813,888
Factor Arrows
213,906,944
Factor Arrows
26,953,472
Factor Arrows
23,476,736
Factor Arrows
21,738,368
Factor Arrows
2869,184
Factor Arrows
2434,592
Factor Arrows
2217,296
Factor Arrows
2108,648
Factor Arrows
254,324
Factor Arrows
227,162
Factor Arrows
213,581
Factor Arrows
34,527
Factor Arrows
31,509
Factor Arrows
3503

More Prime Factorization Examples

111,255,550111,255,551111,255,553111,255,554
21 x 52 x 71 x 411 x 7,7531111 x 10,114,14111031 x 1971 x 5,483121 x 231 x 6611 x 3,6591

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