Q: What is the prime factorization of the number 30,221,505?

 A:
  • The prime factors are: 3 x 3 x 3 x 3 x 5 x 71 x 1,051
    • or also written as { 3, 3, 3, 3, 5, 71, 1,051 }
  • Written in exponential form: 34 x 51 x 711 x 1,0511

Why is the prime factorization of 30,221,505 written as 34 x 51 x 711 x 1,0511?

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 30,221,505

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 30,221,505 by 2

30,221,505 ÷ 2 = 15,110,752.5 - This has a remainder. Let's try another prime number.
30,221,505 ÷ 3 = 10,073,835 - No remainder! 3 is one of the factors!
10,073,835 ÷ 3 = 3,357,945 - No remainder! 3 is one of the factors!
3,357,945 ÷ 3 = 1,119,315 - No remainder! 3 is one of the factors!
1,119,315 ÷ 3 = 373,105 - No remainder! 3 is one of the factors!
373,105 ÷ 3 = 124,368.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
373,105 ÷ 5 = 74,621 - No remainder! 5 is one of the factors!
74,621 ÷ 5 = 14,924.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
74,621 ÷ 7 = 10,660.1429 - This has a remainder. 7 is not a factor.
74,621 ÷ 11 = 6,783.7273 - This has a remainder. 11 is not a factor.
74,621 ÷ 13 = 5,740.0769 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
74,621 ÷ 71 = 1,051 - No remainder! 71 is one of the factors!
1,051 ÷ 71 = 14.8028 - There is a remainder. We can't divide by 71 evenly anymore. Let's try the next prime number
1,051 ÷ 73 = 14.3973 - This has a remainder. 73 is not a factor.
1,051 ÷ 79 = 13.3038 - This has a remainder. 79 is not a factor.
1,051 ÷ 83 = 12.6627 - This has a remainder. 83 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,051 ÷ 1,051 = 1 - No remainder! 1,051 is one of the factors!

The orange divisor(s) above are the prime factors of the number 30,221,505. If we put all of it together we have the factors 3 x 3 x 3 x 3 x 5 x 71 x 1,051 = 30,221,505. It can also be written in exponential form as 34 x 51 x 711 x 1,0511.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 30,221,505.

30,221,505
Factor Arrows
310,073,835
Factor Arrows
33,357,945
Factor Arrows
31,119,315
Factor Arrows
3373,105
Factor Arrows
574,621
Factor Arrows
711,051

More Prime Factorization Examples

30,221,50330,221,50430,221,50630,221,507
131 x 2,324,731126 x 1631 x 2,897121 x 71 x 2,158,679130,221,5071

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