Why is the prime factorization of 124,312,512 written as 26 x 31 x 5871 x 1,1031?
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 124,312,512
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 124,312,512 by 2
124,312,512 ÷ 2 = 62,156,256 - No remainder! 2 is one of the factors!
62,156,256 ÷ 2 = 31,078,128 - No remainder! 2 is one of the factors!
31,078,128 ÷ 2 = 15,539,064 - No remainder! 2 is one of the factors!
15,539,064 ÷ 2 = 7,769,532 - No remainder! 2 is one of the factors!
7,769,532 ÷ 2 = 3,884,766 - No remainder! 2 is one of the factors!
3,884,766 ÷ 2 = 1,942,383 - No remainder! 2 is one of the factors!
1,942,383 ÷ 2 = 971,191.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,942,383 ÷ 3 = 647,461 - No remainder! 3 is one of the factors!
647,461 ÷ 3 = 215,820.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
647,461 ÷ 5 = 129,492.2 - This has a remainder. 5 is not a factor.
647,461 ÷ 7 = 92,494.4286 - This has a remainder. 7 is not a factor.
647,461 ÷ 11 = 58,860.0909 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
647,461 ÷ 587 = 1,103 - No remainder! 587 is one of the factors!
1,103 ÷ 587 = 1.879 - There is a remainder. We can't divide by 587 evenly anymore. Let's try the next prime number
1,103 ÷ 593 = 1.86 - This has a remainder. 593 is not a factor.
1,103 ÷ 599 = 1.8414 - This has a remainder. 599 is not a factor.
1,103 ÷ 601 = 1.8353 - This has a remainder. 601 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,103 ÷ 1,103 = 1 - No remainder! 1,103 is one of the factors!
The orange divisor(s) above are the prime factors of the number 124,312,512. 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 587 x 1,103 = 124,312,512. It can also be written in exponential form as 26 x 31 x 5871 x 1,1031.
Factor Tree
Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 124,312,512.
| 124,312,512 | | | | | | | | |
| ![Factor Arrows](/assets/img/factor-arrows.png) | | | | | | | | |
2 | | 62,156,256 | | | | | | | |
| | ![Factor Arrows](/assets/img/factor-arrows.png) | | | | | | | |
| 2 | | 31,078,128 | | | | | | |
| | | ![Factor Arrows](/assets/img/factor-arrows.png) | | | | | | |
| | 2 | | 15,539,064 | | | | | |
| | | | ![Factor Arrows](/assets/img/factor-arrows.png) | | | | | |
| | | 2 | | 7,769,532 | | | | |
| | | | | ![Factor Arrows](/assets/img/factor-arrows.png) | | | | |
| | | | 2 | | 3,884,766 | | | |
| | | | | | ![Factor Arrows](/assets/img/factor-arrows.png) | | | |
| | | | | 2 | | 1,942,383 | | |
| | | | | | | ![Factor Arrows](/assets/img/factor-arrows.png) | | |
| | | | | | 3 | | 647,461 | |
| | | | | | | | ![Factor Arrows](/assets/img/factor-arrows.png) | |
| | | | | | | 587 | | 1,103 |
More Prime Factorization Examples
124,312,510 | 124,312,511 | 124,312,513 | 124,312,514 |
21 x 51 x 72 x 611 x 4,1591 | 311 x 4,010,0811 | 132 x 8111 x 9071 | 21 x 1991 x 312,3431 |
Try the
factor calculator