Q: What is the prime factorization of the number 84,212,040?

 A:
  • The prime factors are: 2 x 2 x 2 x 3 x 5 x 11 x 131 x 487
    • or also written as { 2, 2, 2, 3, 5, 11, 131, 487 }
  • Written in exponential form: 23 x 31 x 51 x 111 x 1311 x 4871

Why is the prime factorization of 84,212,040 written as 23 x 31 x 51 x 111 x 1311 x 4871?

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 84,212,040

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 84,212,040 by 2

84,212,040 ÷ 2 = 42,106,020 - No remainder! 2 is one of the factors!
42,106,020 ÷ 2 = 21,053,010 - No remainder! 2 is one of the factors!
21,053,010 ÷ 2 = 10,526,505 - No remainder! 2 is one of the factors!
10,526,505 ÷ 2 = 5,263,252.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
10,526,505 ÷ 3 = 3,508,835 - No remainder! 3 is one of the factors!
3,508,835 ÷ 3 = 1,169,611.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
3,508,835 ÷ 5 = 701,767 - No remainder! 5 is one of the factors!
701,767 ÷ 5 = 140,353.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
701,767 ÷ 7 = 100,252.4286 - This has a remainder. 7 is not a factor.
701,767 ÷ 11 = 63,797 - No remainder! 11 is one of the factors!
63,797 ÷ 11 = 5,799.7273 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
63,797 ÷ 13 = 4,907.4615 - This has a remainder. 13 is not a factor.
63,797 ÷ 17 = 3,752.7647 - This has a remainder. 17 is not a factor.
63,797 ÷ 19 = 3,357.7368 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
63,797 ÷ 131 = 487 - No remainder! 131 is one of the factors!
487 ÷ 131 = 3.7176 - There is a remainder. We can't divide by 131 evenly anymore. Let's try the next prime number
487 ÷ 137 = 3.5547 - This has a remainder. 137 is not a factor.
487 ÷ 139 = 3.5036 - This has a remainder. 139 is not a factor.
487 ÷ 149 = 3.2685 - This has a remainder. 149 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
487 ÷ 487 = 1 - No remainder! 487 is one of the factors!

The orange divisor(s) above are the prime factors of the number 84,212,040. If we put all of it together we have the factors 2 x 2 x 2 x 3 x 5 x 11 x 131 x 487 = 84,212,040. It can also be written in exponential form as 23 x 31 x 51 x 111 x 1311 x 4871.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 84,212,040.

84,212,040
Factor Arrows
242,106,020
Factor Arrows
221,053,010
Factor Arrows
210,526,505
Factor Arrows
33,508,835
Factor Arrows
5701,767
Factor Arrows
1163,797
Factor Arrows
131487

More Prime Factorization Examples

84,212,03884,212,03984,212,04184,212,042
21 x 1391 x 302,9211232 x 159,191184,212,041121 x 3471 x 121,3431

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