Q: What is the prime factorization of the number 21,618,240?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 7 x 3,217
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 5, 7, 3,217 }
  • Written in exponential form: 26 x 31 x 51 x 71 x 3,2171

Why is the prime factorization of 21,618,240 written as 26 x 31 x 51 x 71 x 3,2171?

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 21,618,240

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 21,618,240 by 2

21,618,240 ÷ 2 = 10,809,120 - No remainder! 2 is one of the factors!
10,809,120 ÷ 2 = 5,404,560 - No remainder! 2 is one of the factors!
5,404,560 ÷ 2 = 2,702,280 - No remainder! 2 is one of the factors!
2,702,280 ÷ 2 = 1,351,140 - No remainder! 2 is one of the factors!
1,351,140 ÷ 2 = 675,570 - No remainder! 2 is one of the factors!
675,570 ÷ 2 = 337,785 - No remainder! 2 is one of the factors!
337,785 ÷ 2 = 168,892.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
337,785 ÷ 3 = 112,595 - No remainder! 3 is one of the factors!
112,595 ÷ 3 = 37,531.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
112,595 ÷ 5 = 22,519 - No remainder! 5 is one of the factors!
22,519 ÷ 5 = 4,503.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
22,519 ÷ 7 = 3,217 - No remainder! 7 is one of the factors!
3,217 ÷ 7 = 459.5714 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
3,217 ÷ 11 = 292.4545 - This has a remainder. 11 is not a factor.
3,217 ÷ 13 = 247.4615 - This has a remainder. 13 is not a factor.
3,217 ÷ 17 = 189.2353 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
3,217 ÷ 3,217 = 1 - No remainder! 3,217 is one of the factors!

The orange divisor(s) above are the prime factors of the number 21,618,240. 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 5 x 7 x 3,217 = 21,618,240. It can also be written in exponential form as 26 x 31 x 51 x 71 x 3,2171.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 21,618,240.

21,618,240
Factor Arrows
210,809,120
Factor Arrows
25,404,560
Factor Arrows
22,702,280
Factor Arrows
21,351,140
Factor Arrows
2675,570
Factor Arrows
2337,785
Factor Arrows
3112,595
Factor Arrows
522,519
Factor Arrows
73,217

More Prime Factorization Examples

21,618,23821,618,23921,618,24121,618,242
21 x 191 x 3971 x 1,43314911 x 44,029121,618,241121 x 1011 x 107,0211

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