Q: What is the prime factorization of the number 76,812,800?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 17 x 353
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 17, 353 }
  • Written in exponential form: 29 x 52 x 171 x 3531

Why is the prime factorization of 76,812,800 written as 29 x 52 x 171 x 3531?

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 76,812,800

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 76,812,800 by 2

76,812,800 ÷ 2 = 38,406,400 - No remainder! 2 is one of the factors!
38,406,400 ÷ 2 = 19,203,200 - No remainder! 2 is one of the factors!
19,203,200 ÷ 2 = 9,601,600 - No remainder! 2 is one of the factors!
9,601,600 ÷ 2 = 4,800,800 - No remainder! 2 is one of the factors!
4,800,800 ÷ 2 = 2,400,400 - No remainder! 2 is one of the factors!
2,400,400 ÷ 2 = 1,200,200 - No remainder! 2 is one of the factors!
1,200,200 ÷ 2 = 600,100 - No remainder! 2 is one of the factors!
600,100 ÷ 2 = 300,050 - No remainder! 2 is one of the factors!
300,050 ÷ 2 = 150,025 - No remainder! 2 is one of the factors!
150,025 ÷ 2 = 75,012.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
150,025 ÷ 3 = 50,008.3333 - This has a remainder. 3 is not a factor.
150,025 ÷ 5 = 30,005 - No remainder! 5 is one of the factors!
30,005 ÷ 5 = 6,001 - No remainder! 5 is one of the factors!
6,001 ÷ 5 = 1,200.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
6,001 ÷ 7 = 857.2857 - This has a remainder. 7 is not a factor.
6,001 ÷ 11 = 545.5455 - This has a remainder. 11 is not a factor.
6,001 ÷ 13 = 461.6154 - This has a remainder. 13 is not a factor.
6,001 ÷ 17 = 353 - No remainder! 17 is one of the factors!
353 ÷ 17 = 20.7647 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
353 ÷ 19 = 18.5789 - This has a remainder. 19 is not a factor.
353 ÷ 23 = 15.3478 - This has a remainder. 23 is not a factor.
353 ÷ 29 = 12.1724 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
353 ÷ 353 = 1 - No remainder! 353 is one of the factors!

The orange divisor(s) above are the prime factors of the number 76,812,800. 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 2 x 5 x 5 x 17 x 353 = 76,812,800. It can also be written in exponential form as 29 x 52 x 171 x 3531.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 76,812,800.

76,812,800
Factor Arrows
238,406,400
Factor Arrows
219,203,200
Factor Arrows
29,601,600
Factor Arrows
24,800,800
Factor Arrows
22,400,400
Factor Arrows
21,200,200
Factor Arrows
2600,100
Factor Arrows
2300,050
Factor Arrows
2150,025
Factor Arrows
530,005
Factor Arrows
56,001
Factor Arrows
17353

More Prime Factorization Examples

76,812,79876,812,79976,812,80176,812,802
21 x 31 x 6171 x 20,749171 x 1,2891 x 8,513131 x 131 x 191 x 231 x 4,507121 x 111 x 3,491,4911

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