Q: What is the prime factorization of the number 1,365,200?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 5 x 5 x 3,413
    • or also written as { 2, 2, 2, 2, 5, 5, 3,413 }
  • Written in exponential form: 24 x 52 x 3,4131

Why is the prime factorization of 1,365,200 written as 24 x 52 x 3,4131?

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 1,365,200

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 1,365,200 by 2

1,365,200 ÷ 2 = 682,600 - No remainder! 2 is one of the factors!
682,600 ÷ 2 = 341,300 - No remainder! 2 is one of the factors!
341,300 ÷ 2 = 170,650 - No remainder! 2 is one of the factors!
170,650 ÷ 2 = 85,325 - No remainder! 2 is one of the factors!
85,325 ÷ 2 = 42,662.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
85,325 ÷ 3 = 28,441.6667 - This has a remainder. 3 is not a factor.
85,325 ÷ 5 = 17,065 - No remainder! 5 is one of the factors!
17,065 ÷ 5 = 3,413 - No remainder! 5 is one of the factors!
3,413 ÷ 5 = 682.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
3,413 ÷ 7 = 487.5714 - This has a remainder. 7 is not a factor.
3,413 ÷ 11 = 310.2727 - This has a remainder. 11 is not a factor.
3,413 ÷ 13 = 262.5385 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
3,413 ÷ 3,413 = 1 - No remainder! 3,413 is one of the factors!

The orange divisor(s) above are the prime factors of the number 1,365,200. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 5 x 5 x 3,413 = 1,365,200. It can also be written in exponential form as 24 x 52 x 3,4131.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 1,365,200.

1,365,200
Factor Arrows
2682,600
Factor Arrows
2341,300
Factor Arrows
2170,650
Factor Arrows
285,325
Factor Arrows
517,065
Factor Arrows
53,413

More Prime Factorization Examples

1,365,1981,365,1991,365,2011,365,202
21 x 31 x 227,5331111 x 791 x 1,571133 x 591 x 857121 x 171 x 40,1531

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