Q: What is the prime factorization of the number 535,365,609?

 A:
  • The prime factors are: 3 x 43 x 1,093 x 3,797
    • or also written as { 3, 43, 1,093, 3,797 }
  • Written in exponential form: 31 x 431 x 1,0931 x 3,7971

Why is the prime factorization of 535,365,609 written as 31 x 431 x 1,0931 x 3,7971?

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 535,365,609

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 535,365,609 by 2

535,365,609 ÷ 2 = 267,682,804.5 - This has a remainder. Let's try another prime number.
535,365,609 ÷ 3 = 178,455,203 - No remainder! 3 is one of the factors!
178,455,203 ÷ 3 = 59,485,067.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
178,455,203 ÷ 5 = 35,691,040.6 - This has a remainder. 5 is not a factor.
178,455,203 ÷ 7 = 25,493,600.4286 - This has a remainder. 7 is not a factor.
178,455,203 ÷ 11 = 16,223,200.2727 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
178,455,203 ÷ 43 = 4,150,121 - No remainder! 43 is one of the factors!
4,150,121 ÷ 43 = 96,514.4419 - There is a remainder. We can't divide by 43 evenly anymore. Let's try the next prime number
4,150,121 ÷ 47 = 88,300.4468 - This has a remainder. 47 is not a factor.
4,150,121 ÷ 53 = 78,304.1698 - This has a remainder. 53 is not a factor.
4,150,121 ÷ 59 = 70,341.0339 - This has a remainder. 59 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
4,150,121 ÷ 1,093 = 3,797 - No remainder! 1,093 is one of the factors!
3,797 ÷ 1,093 = 3.4739 - There is a remainder. We can't divide by 1093 evenly anymore. Let's try the next prime number
3,797 ÷ 1,097 = 3.4613 - This has a remainder. 1,097 is not a factor.
3,797 ÷ 1,103 = 3.4424 - This has a remainder. 1,103 is not a factor.
3,797 ÷ 1,109 = 3.4238 - This has a remainder. 1,109 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
3,797 ÷ 3,797 = 1 - No remainder! 3,797 is one of the factors!

The orange divisor(s) above are the prime factors of the number 535,365,609. If we put all of it together we have the factors 3 x 43 x 1,093 x 3,797 = 535,365,609. It can also be written in exponential form as 31 x 431 x 1,0931 x 3,7971.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 535,365,609.

535,365,609
Factor Arrows
3178,455,203
Factor Arrows
434,150,121
Factor Arrows
1,0933,797

More Prime Factorization Examples

535,365,607535,365,608535,365,610535,365,611
71 x 291 x 2,637,269123 x 7331 x 91,297121 x 51 x 131 x 4,118,1971111 x 48,669,6011

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