Q: What is the prime factorization of the number 5,443,200?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 5 x 5 x 7
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 7 }
  • Written in exponential form: 27 x 35 x 52 x 71

Why is the prime factorization of 5,443,200 written as 27 x 35 x 52 x 71?

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 5,443,200

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 5,443,200 by 2

5,443,200 ÷ 2 = 2,721,600 - No remainder! 2 is one of the factors!
2,721,600 ÷ 2 = 1,360,800 - No remainder! 2 is one of the factors!
1,360,800 ÷ 2 = 680,400 - No remainder! 2 is one of the factors!
680,400 ÷ 2 = 340,200 - No remainder! 2 is one of the factors!
340,200 ÷ 2 = 170,100 - No remainder! 2 is one of the factors!
170,100 ÷ 2 = 85,050 - No remainder! 2 is one of the factors!
85,050 ÷ 2 = 42,525 - No remainder! 2 is one of the factors!
42,525 ÷ 2 = 21,262.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
42,525 ÷ 3 = 14,175 - No remainder! 3 is one of the factors!
14,175 ÷ 3 = 4,725 - No remainder! 3 is one of the factors!
4,725 ÷ 3 = 1,575 - No remainder! 3 is one of the factors!
1,575 ÷ 3 = 525 - No remainder! 3 is one of the factors!
525 ÷ 3 = 175 - No remainder! 3 is one of the factors!
175 ÷ 3 = 58.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
175 ÷ 5 = 35 - No remainder! 5 is one of the factors!
35 ÷ 5 = 7 - No remainder! 5 is one of the factors!
7 ÷ 5 = 1.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
7 ÷ 7 = 1 - No remainder! 7 is one of the factors!

The orange divisor(s) above are the prime factors of the number 5,443,200. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 5 x 5 x 7 = 5,443,200. It can also be written in exponential form as 27 x 35 x 52 x 71.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 5,443,200.

5,443,200
Factor Arrows
22,721,600
Factor Arrows
21,360,800
Factor Arrows
2680,400
Factor Arrows
2340,200
Factor Arrows
2170,100
Factor Arrows
285,050
Factor Arrows
242,525
Factor Arrows
314,175
Factor Arrows
34,725
Factor Arrows
31,575
Factor Arrows
3525
Factor Arrows
3175
Factor Arrows
535
Factor Arrows
57

More Prime Factorization Examples

5,443,1985,443,1995,443,2015,443,202
21 x 431 x 1671 x 37914871 x 11,1771411 x 132,761121 x 1,5231 x 1,7871

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