Why is the prime factorization of 712,535,600 written as 24 x 52 x 71 x 3971 x 6411?
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 712,535,600
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 712,535,600 by 2
712,535,600 ÷ 2 = 356,267,800 - No remainder! 2 is one of the factors!
356,267,800 ÷ 2 = 178,133,900 - No remainder! 2 is one of the factors!
178,133,900 ÷ 2 = 89,066,950 - No remainder! 2 is one of the factors!
89,066,950 ÷ 2 = 44,533,475 - No remainder! 2 is one of the factors!
44,533,475 ÷ 2 = 22,266,737.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
44,533,475 ÷ 3 = 14,844,491.6667 - This has a remainder. 3 is not a factor.
44,533,475 ÷ 5 = 8,906,695 - No remainder! 5 is one of the factors!
8,906,695 ÷ 5 = 1,781,339 - No remainder! 5 is one of the factors!
1,781,339 ÷ 5 = 356,267.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
1,781,339 ÷ 7 = 254,477 - No remainder! 7 is one of the factors!
254,477 ÷ 7 = 36,353.8571 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
254,477 ÷ 11 = 23,134.2727 - This has a remainder. 11 is not a factor.
254,477 ÷ 13 = 19,575.1538 - This has a remainder. 13 is not a factor.
254,477 ÷ 17 = 14,969.2353 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
254,477 ÷ 397 = 641 - No remainder! 397 is one of the factors!
641 ÷ 397 = 1.6146 - There is a remainder. We can't divide by 397 evenly anymore. Let's try the next prime number
641 ÷ 401 = 1.5985 - This has a remainder. 401 is not a factor.
641 ÷ 409 = 1.5672 - This has a remainder. 409 is not a factor.
641 ÷ 419 = 1.5298 - This has a remainder. 419 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
641 ÷ 641 = 1 - No remainder! 641 is one of the factors!
The orange divisor(s) above are the prime factors of the number 712,535,600. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 5 x 5 x 7 x 397 x 641 = 712,535,600. It can also be written in exponential form as 24 x 52 x 71 x 3971 x 6411.
Factor Tree
Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 712,535,600.
| 712,535,600 | | | | | | | | |
| ![Factor Arrows](/assets/img/factor-arrows.png) | | | | | | | | |
2 | | 356,267,800 | | | | | | | |
| | ![Factor Arrows](/assets/img/factor-arrows.png) | | | | | | | |
| 2 | | 178,133,900 | | | | | | |
| | | ![Factor Arrows](/assets/img/factor-arrows.png) | | | | | | |
| | 2 | | 89,066,950 | | | | | |
| | | | ![Factor Arrows](/assets/img/factor-arrows.png) | | | | | |
| | | 2 | | 44,533,475 | | | | |
| | | | | ![Factor Arrows](/assets/img/factor-arrows.png) | | | | |
| | | | 5 | | 8,906,695 | | | |
| | | | | | ![Factor Arrows](/assets/img/factor-arrows.png) | | | |
| | | | | 5 | | 1,781,339 | | |
| | | | | | | ![Factor Arrows](/assets/img/factor-arrows.png) | | |
| | | | | | 7 | | 254,477 | |
| | | | | | | | ![Factor Arrows](/assets/img/factor-arrows.png) | |
| | | | | | | 397 | | 641 |
More Prime Factorization Examples
712,535,598 | 712,535,599 | 712,535,601 | 712,535,602 |
21 x 32 x 2,4731 x 16,0071 | 712,535,5991 | 31 x 10,0911 x 23,5371 | 21 x 791 x 891 x 50,6711 |
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