Q: What is the prime factorization of the number 601,232,064?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 17 x 167 x 1,103
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 17, 167, 1,103 }
  • Written in exponential form: 26 x 31 x 171 x 1671 x 1,1031

Why is the prime factorization of 601,232,064 written as 26 x 31 x 171 x 1671 x 1,1031?

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 601,232,064

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 601,232,064 by 2

601,232,064 ÷ 2 = 300,616,032 - No remainder! 2 is one of the factors!
300,616,032 ÷ 2 = 150,308,016 - No remainder! 2 is one of the factors!
150,308,016 ÷ 2 = 75,154,008 - No remainder! 2 is one of the factors!
75,154,008 ÷ 2 = 37,577,004 - No remainder! 2 is one of the factors!
37,577,004 ÷ 2 = 18,788,502 - No remainder! 2 is one of the factors!
18,788,502 ÷ 2 = 9,394,251 - No remainder! 2 is one of the factors!
9,394,251 ÷ 2 = 4,697,125.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
9,394,251 ÷ 3 = 3,131,417 - No remainder! 3 is one of the factors!
3,131,417 ÷ 3 = 1,043,805.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
3,131,417 ÷ 5 = 626,283.4 - This has a remainder. 5 is not a factor.
3,131,417 ÷ 7 = 447,345.2857 - This has a remainder. 7 is not a factor.
3,131,417 ÷ 11 = 284,674.2727 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
3,131,417 ÷ 17 = 184,201 - No remainder! 17 is one of the factors!
184,201 ÷ 17 = 10,835.3529 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
184,201 ÷ 19 = 9,694.7895 - This has a remainder. 19 is not a factor.
184,201 ÷ 23 = 8,008.7391 - This has a remainder. 23 is not a factor.
184,201 ÷ 29 = 6,351.7586 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
184,201 ÷ 167 = 1,103 - No remainder! 167 is one of the factors!
1,103 ÷ 167 = 6.6048 - There is a remainder. We can't divide by 167 evenly anymore. Let's try the next prime number
1,103 ÷ 173 = 6.3757 - This has a remainder. 173 is not a factor.
1,103 ÷ 179 = 6.162 - This has a remainder. 179 is not a factor.
1,103 ÷ 181 = 6.0939 - This has a remainder. 181 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,103 ÷ 1,103 = 1 - No remainder! 1,103 is one of the factors!

The orange divisor(s) above are the prime factors of the number 601,232,064. 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 17 x 167 x 1,103 = 601,232,064. It can also be written in exponential form as 26 x 31 x 171 x 1671 x 1,1031.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 601,232,064.

601,232,064
Factor Arrows
2300,616,032
Factor Arrows
2150,308,016
Factor Arrows
275,154,008
Factor Arrows
237,577,004
Factor Arrows
218,788,502
Factor Arrows
29,394,251
Factor Arrows
33,131,417
Factor Arrows
17184,201
Factor Arrows
1671,103

More Prime Factorization Examples

601,232,062601,232,063601,232,065601,232,066
21 x 300,616,0311431 x 13,982,141151 x 71 x 17,178,059121 x 591 x 5,095,1871

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