Q: What is the prime factorization of the number 12,798,570?

 A:
  • The prime factors are: 2 x 3 x 5 x 29 x 47 x 313
    • or also written as { 2, 3, 5, 29, 47, 313 }
  • Written in exponential form: 21 x 31 x 51 x 291 x 471 x 3131

Why is the prime factorization of 12,798,570 written as 21 x 31 x 51 x 291 x 471 x 3131?

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 12,798,570

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 12,798,570 by 2

12,798,570 ÷ 2 = 6,399,285 - No remainder! 2 is one of the factors!
6,399,285 ÷ 2 = 3,199,642.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
6,399,285 ÷ 3 = 2,133,095 - No remainder! 3 is one of the factors!
2,133,095 ÷ 3 = 711,031.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
2,133,095 ÷ 5 = 426,619 - No remainder! 5 is one of the factors!
426,619 ÷ 5 = 85,323.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
426,619 ÷ 7 = 60,945.5714 - This has a remainder. 7 is not a factor.
426,619 ÷ 11 = 38,783.5455 - This has a remainder. 11 is not a factor.
426,619 ÷ 13 = 32,816.8462 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
426,619 ÷ 29 = 14,711 - No remainder! 29 is one of the factors!
14,711 ÷ 29 = 507.2759 - There is a remainder. We can't divide by 29 evenly anymore. Let's try the next prime number
14,711 ÷ 31 = 474.5484 - This has a remainder. 31 is not a factor.
14,711 ÷ 37 = 397.5946 - This has a remainder. 37 is not a factor.
14,711 ÷ 41 = 358.8049 - This has a remainder. 41 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
14,711 ÷ 47 = 313 - No remainder! 47 is one of the factors!
313 ÷ 47 = 6.6596 - There is a remainder. We can't divide by 47 evenly anymore. Let's try the next prime number
313 ÷ 53 = 5.9057 - This has a remainder. 53 is not a factor.
313 ÷ 59 = 5.3051 - This has a remainder. 59 is not a factor.
313 ÷ 61 = 5.1311 - This has a remainder. 61 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
313 ÷ 313 = 1 - No remainder! 313 is one of the factors!

The orange divisor(s) above are the prime factors of the number 12,798,570. If we put all of it together we have the factors 2 x 3 x 5 x 29 x 47 x 313 = 12,798,570. It can also be written in exponential form as 21 x 31 x 51 x 291 x 471 x 3131.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 12,798,570.

12,798,570
Factor Arrows
26,399,285
Factor Arrows
32,133,095
Factor Arrows
5426,619
Factor Arrows
2914,711
Factor Arrows
47313

More Prime Factorization Examples

12,798,56812,798,56912,798,57112,798,572
23 x 971 x 16,493171 x 171 x 1311 x 8211191 x 673,609122 x 3,199,6431

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