Q: What is the prime factorization of the number 120,286,020?

 A:
  • The prime factors are: 2 x 2 x 3 x 5 x 1,399 x 1,433
    • or also written as { 2, 2, 3, 5, 1,399, 1,433 }
  • Written in exponential form: 22 x 31 x 51 x 1,3991 x 1,4331

Why is the prime factorization of 120,286,020 written as 22 x 31 x 51 x 1,3991 x 1,4331?

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 120,286,020

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 120,286,020 by 2

120,286,020 ÷ 2 = 60,143,010 - No remainder! 2 is one of the factors!
60,143,010 ÷ 2 = 30,071,505 - No remainder! 2 is one of the factors!
30,071,505 ÷ 2 = 15,035,752.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
30,071,505 ÷ 3 = 10,023,835 - No remainder! 3 is one of the factors!
10,023,835 ÷ 3 = 3,341,278.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
10,023,835 ÷ 5 = 2,004,767 - No remainder! 5 is one of the factors!
2,004,767 ÷ 5 = 400,953.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
2,004,767 ÷ 7 = 286,395.2857 - This has a remainder. 7 is not a factor.
2,004,767 ÷ 11 = 182,251.5455 - This has a remainder. 11 is not a factor.
2,004,767 ÷ 13 = 154,212.8462 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,004,767 ÷ 1,399 = 1,433 - No remainder! 1,399 is one of the factors!
1,433 ÷ 1,399 = 1.0243 - There is a remainder. We can't divide by 1399 evenly anymore. Let's try the next prime number
1,433 ÷ 1,409 = 1.017 - This has a remainder. 1,409 is not a factor.
1,433 ÷ 1,423 = 1.007 - This has a remainder. 1,423 is not a factor.
1,433 ÷ 1,427 = 1.0042 - This has a remainder. 1,427 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,433 ÷ 1,433 = 1 - No remainder! 1,433 is one of the factors!

The orange divisor(s) above are the prime factors of the number 120,286,020. If we put all of it together we have the factors 2 x 2 x 3 x 5 x 1,399 x 1,433 = 120,286,020. It can also be written in exponential form as 22 x 31 x 51 x 1,3991 x 1,4331.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 120,286,020.

120,286,020
Factor Arrows
260,143,010
Factor Arrows
230,071,505
Factor Arrows
310,023,835
Factor Arrows
52,004,767
Factor Arrows
1,3991,433

More Prime Factorization Examples

120,286,018120,286,019120,286,021120,286,022
21 x 1271 x 4431 x 1,069171 x 471 x 365,6111231 x 1571 x 33,311121 x 611 x 985,9511

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