Q: What is the prime factorization of the number 21,221,430?

 A:
  • The prime factors are: 2 x 3 x 5 x 229 x 3,089
    • or also written as { 2, 3, 5, 229, 3,089 }
  • Written in exponential form: 21 x 31 x 51 x 2291 x 3,0891

Why is the prime factorization of 21,221,430 written as 21 x 31 x 51 x 2291 x 3,0891?

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 21,221,430

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 21,221,430 by 2

21,221,430 ÷ 2 = 10,610,715 - No remainder! 2 is one of the factors!
10,610,715 ÷ 2 = 5,305,357.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
10,610,715 ÷ 3 = 3,536,905 - No remainder! 3 is one of the factors!
3,536,905 ÷ 3 = 1,178,968.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
3,536,905 ÷ 5 = 707,381 - No remainder! 5 is one of the factors!
707,381 ÷ 5 = 141,476.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
707,381 ÷ 7 = 101,054.4286 - This has a remainder. 7 is not a factor.
707,381 ÷ 11 = 64,307.3636 - This has a remainder. 11 is not a factor.
707,381 ÷ 13 = 54,413.9231 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
707,381 ÷ 229 = 3,089 - No remainder! 229 is one of the factors!
3,089 ÷ 229 = 13.4891 - There is a remainder. We can't divide by 229 evenly anymore. Let's try the next prime number
3,089 ÷ 233 = 13.2575 - This has a remainder. 233 is not a factor.
3,089 ÷ 239 = 12.9247 - This has a remainder. 239 is not a factor.
3,089 ÷ 241 = 12.8174 - This has a remainder. 241 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
3,089 ÷ 3,089 = 1 - No remainder! 3,089 is one of the factors!

The orange divisor(s) above are the prime factors of the number 21,221,430. If we put all of it together we have the factors 2 x 3 x 5 x 229 x 3,089 = 21,221,430. It can also be written in exponential form as 21 x 31 x 51 x 2291 x 3,0891.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 21,221,430.

21,221,430
Factor Arrows
210,610,715
Factor Arrows
33,536,905
Factor Arrows
5707,381
Factor Arrows
2293,089

More Prime Factorization Examples

21,221,42821,221,42921,221,43121,221,432
22 x 1091 x 48,673121,221,429171 x 111 x 1971 x 1,399123 x 9411 x 2,8191

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