Q: What is the prime factorization of the number 19,229,175?

 A:
  • The prime factors are: 3 x 3 x 5 x 5 x 7 x 29 x 421
    • or also written as { 3, 3, 5, 5, 7, 29, 421 }
  • Written in exponential form: 32 x 52 x 71 x 291 x 4211

Why is the prime factorization of 19,229,175 written as 32 x 52 x 71 x 291 x 4211?

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 19,229,175

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 19,229,175 by 2

19,229,175 ÷ 2 = 9,614,587.5 - This has a remainder. Let's try another prime number.
19,229,175 ÷ 3 = 6,409,725 - No remainder! 3 is one of the factors!
6,409,725 ÷ 3 = 2,136,575 - No remainder! 3 is one of the factors!
2,136,575 ÷ 3 = 712,191.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
2,136,575 ÷ 5 = 427,315 - No remainder! 5 is one of the factors!
427,315 ÷ 5 = 85,463 - No remainder! 5 is one of the factors!
85,463 ÷ 5 = 17,092.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
85,463 ÷ 7 = 12,209 - No remainder! 7 is one of the factors!
12,209 ÷ 7 = 1,744.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
12,209 ÷ 11 = 1,109.9091 - This has a remainder. 11 is not a factor.
12,209 ÷ 13 = 939.1538 - This has a remainder. 13 is not a factor.
12,209 ÷ 17 = 718.1765 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
12,209 ÷ 29 = 421 - No remainder! 29 is one of the factors!
421 ÷ 29 = 14.5172 - There is a remainder. We can't divide by 29 evenly anymore. Let's try the next prime number
421 ÷ 31 = 13.5806 - This has a remainder. 31 is not a factor.
421 ÷ 37 = 11.3784 - This has a remainder. 37 is not a factor.
421 ÷ 41 = 10.2683 - This has a remainder. 41 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
421 ÷ 421 = 1 - No remainder! 421 is one of the factors!

The orange divisor(s) above are the prime factors of the number 19,229,175. If we put all of it together we have the factors 3 x 3 x 5 x 5 x 7 x 29 x 421 = 19,229,175. It can also be written in exponential form as 32 x 52 x 71 x 291 x 4211.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 19,229,175.

19,229,175
Factor Arrows
36,409,725
Factor Arrows
32,136,575
Factor Arrows
5427,315
Factor Arrows
585,463
Factor Arrows
712,209
Factor Arrows
29421

More Prime Factorization Examples

19,229,17319,229,17419,229,17619,229,177
231 x 1031 x 8,117121 x 9,614,587123 x 171 x 311 x 4,5611111 x 1,748,1071

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