Q: What is the prime factorization of the number 104,212,416?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 7 x 7 x 11 x 19 x 53
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 7, 7, 11, 19, 53 }
  • Written in exponential form: 26 x 31 x 72 x 111 x 191 x 531

Why is the prime factorization of 104,212,416 written as 26 x 31 x 72 x 111 x 191 x 531?

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 104,212,416

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 104,212,416 by 2

104,212,416 ÷ 2 = 52,106,208 - No remainder! 2 is one of the factors!
52,106,208 ÷ 2 = 26,053,104 - No remainder! 2 is one of the factors!
26,053,104 ÷ 2 = 13,026,552 - No remainder! 2 is one of the factors!
13,026,552 ÷ 2 = 6,513,276 - No remainder! 2 is one of the factors!
6,513,276 ÷ 2 = 3,256,638 - No remainder! 2 is one of the factors!
3,256,638 ÷ 2 = 1,628,319 - No remainder! 2 is one of the factors!
1,628,319 ÷ 2 = 814,159.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,628,319 ÷ 3 = 542,773 - No remainder! 3 is one of the factors!
542,773 ÷ 3 = 180,924.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
542,773 ÷ 5 = 108,554.6 - This has a remainder. 5 is not a factor.
542,773 ÷ 7 = 77,539 - No remainder! 7 is one of the factors!
77,539 ÷ 7 = 11,077 - No remainder! 7 is one of the factors!
11,077 ÷ 7 = 1,582.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
11,077 ÷ 11 = 1,007 - No remainder! 11 is one of the factors!
1,007 ÷ 11 = 91.5455 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
1,007 ÷ 13 = 77.4615 - This has a remainder. 13 is not a factor.
1,007 ÷ 17 = 59.2353 - This has a remainder. 17 is not a factor.
1,007 ÷ 19 = 53 - No remainder! 19 is one of the factors!
53 ÷ 19 = 2.7895 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
53 ÷ 23 = 2.3043 - This has a remainder. 23 is not a factor.
53 ÷ 29 = 1.8276 - This has a remainder. 29 is not a factor.
53 ÷ 31 = 1.7097 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
53 ÷ 53 = 1 - No remainder! 53 is one of the factors!

The orange divisor(s) above are the prime factors of the number 104,212,416. 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 7 x 7 x 11 x 19 x 53 = 104,212,416. It can also be written in exponential form as 26 x 31 x 72 x 111 x 191 x 531.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 104,212,416.

104,212,416
Factor Arrows
252,106,208
Factor Arrows
226,053,104
Factor Arrows
213,026,552
Factor Arrows
26,513,276
Factor Arrows
23,256,638
Factor Arrows
21,628,319
Factor Arrows
3542,773
Factor Arrows
777,539
Factor Arrows
711,077
Factor Arrows
111,007
Factor Arrows
1953

More Prime Factorization Examples

104,212,414104,212,415104,212,417104,212,418
21 x 171 x 891 x 34,439151 x 20,842,4831104,212,417121 x 52,106,2091

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