Q: What is the prime factorization of the number 35,320,064?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 13 x 10,613
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 13, 10,613 }
  • Written in exponential form: 28 x 131 x 10,6131

Why is the prime factorization of 35,320,064 written as 28 x 131 x 10,6131?

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 35,320,064

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 35,320,064 by 2

35,320,064 ÷ 2 = 17,660,032 - No remainder! 2 is one of the factors!
17,660,032 ÷ 2 = 8,830,016 - No remainder! 2 is one of the factors!
8,830,016 ÷ 2 = 4,415,008 - No remainder! 2 is one of the factors!
4,415,008 ÷ 2 = 2,207,504 - No remainder! 2 is one of the factors!
2,207,504 ÷ 2 = 1,103,752 - No remainder! 2 is one of the factors!
1,103,752 ÷ 2 = 551,876 - No remainder! 2 is one of the factors!
551,876 ÷ 2 = 275,938 - No remainder! 2 is one of the factors!
275,938 ÷ 2 = 137,969 - No remainder! 2 is one of the factors!
137,969 ÷ 2 = 68,984.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
137,969 ÷ 3 = 45,989.6667 - This has a remainder. 3 is not a factor.
137,969 ÷ 5 = 27,593.8 - This has a remainder. 5 is not a factor.
137,969 ÷ 7 = 19,709.8571 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
137,969 ÷ 13 = 10,613 - No remainder! 13 is one of the factors!
10,613 ÷ 13 = 816.3846 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
10,613 ÷ 17 = 624.2941 - This has a remainder. 17 is not a factor.
10,613 ÷ 19 = 558.5789 - This has a remainder. 19 is not a factor.
10,613 ÷ 23 = 461.4348 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
10,613 ÷ 10,613 = 1 - No remainder! 10,613 is one of the factors!

The orange divisor(s) above are the prime factors of the number 35,320,064. 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 2 x 13 x 10,613 = 35,320,064. It can also be written in exponential form as 28 x 131 x 10,6131.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 35,320,064.

35,320,064
Factor Arrows
217,660,032
Factor Arrows
28,830,016
Factor Arrows
24,415,008
Factor Arrows
22,207,504
Factor Arrows
21,103,752
Factor Arrows
2551,876
Factor Arrows
2275,938
Factor Arrows
2137,969
Factor Arrows
1310,613

More Prime Factorization Examples

35,320,06235,320,06335,320,06535,320,066
21 x 31 x 5,886,677135,320,063131 x 51 x 111 x 231 x 411 x 227121 x 17,660,0331

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