Q: What is the prime factorization of the number 520,560?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 241
    • or also written as { 2, 2, 2, 2, 3, 3, 3, 5, 241 }
  • Written in exponential form: 24 x 33 x 51 x 2411

Why is the prime factorization of 520,560 written as 24 x 33 x 51 x 2411?

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 520,560

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 520,560 by 2

520,560 ÷ 2 = 260,280 - No remainder! 2 is one of the factors!
260,280 ÷ 2 = 130,140 - No remainder! 2 is one of the factors!
130,140 ÷ 2 = 65,070 - No remainder! 2 is one of the factors!
65,070 ÷ 2 = 32,535 - No remainder! 2 is one of the factors!
32,535 ÷ 2 = 16,267.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
32,535 ÷ 3 = 10,845 - No remainder! 3 is one of the factors!
10,845 ÷ 3 = 3,615 - No remainder! 3 is one of the factors!
3,615 ÷ 3 = 1,205 - No remainder! 3 is one of the factors!
1,205 ÷ 3 = 401.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
1,205 ÷ 5 = 241 - No remainder! 5 is one of the factors!
241 ÷ 5 = 48.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
241 ÷ 7 = 34.4286 - This has a remainder. 7 is not a factor.
241 ÷ 11 = 21.9091 - This has a remainder. 11 is not a factor.
241 ÷ 13 = 18.5385 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
241 ÷ 241 = 1 - No remainder! 241 is one of the factors!

The orange divisor(s) above are the prime factors of the number 520,560. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 241 = 520,560. It can also be written in exponential form as 24 x 33 x 51 x 2411.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 520,560.

520,560
Factor Arrows
2260,280
Factor Arrows
2130,140
Factor Arrows
265,070
Factor Arrows
232,535
Factor Arrows
310,845
Factor Arrows
33,615
Factor Arrows
31,205
Factor Arrows
5241

More Prime Factorization Examples

520,558520,559520,561520,562
21 x 431 x 6,0531131 x 231 x 1,7411891 x 5,849121 x 71 x 192 x 1031

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