Q: What is the prime factorization of the number 3,414,224?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 11 x 19 x 1,021
    • or also written as { 2, 2, 2, 2, 11, 19, 1,021 }
  • Written in exponential form: 24 x 111 x 191 x 1,0211

Why is the prime factorization of 3,414,224 written as 24 x 111 x 191 x 1,0211?

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 3,414,224

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 3,414,224 by 2

3,414,224 ÷ 2 = 1,707,112 - No remainder! 2 is one of the factors!
1,707,112 ÷ 2 = 853,556 - No remainder! 2 is one of the factors!
853,556 ÷ 2 = 426,778 - No remainder! 2 is one of the factors!
426,778 ÷ 2 = 213,389 - No remainder! 2 is one of the factors!
213,389 ÷ 2 = 106,694.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
213,389 ÷ 3 = 71,129.6667 - This has a remainder. 3 is not a factor.
213,389 ÷ 5 = 42,677.8 - This has a remainder. 5 is not a factor.
213,389 ÷ 7 = 30,484.1429 - This has a remainder. 7 is not a factor.
213,389 ÷ 11 = 19,399 - No remainder! 11 is one of the factors!
19,399 ÷ 11 = 1,763.5455 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
19,399 ÷ 13 = 1,492.2308 - This has a remainder. 13 is not a factor.
19,399 ÷ 17 = 1,141.1176 - This has a remainder. 17 is not a factor.
19,399 ÷ 19 = 1,021 - No remainder! 19 is one of the factors!
1,021 ÷ 19 = 53.7368 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
1,021 ÷ 23 = 44.3913 - This has a remainder. 23 is not a factor.
1,021 ÷ 29 = 35.2069 - This has a remainder. 29 is not a factor.
1,021 ÷ 31 = 32.9355 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,021 ÷ 1,021 = 1 - No remainder! 1,021 is one of the factors!

The orange divisor(s) above are the prime factors of the number 3,414,224. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 11 x 19 x 1,021 = 3,414,224. It can also be written in exponential form as 24 x 111 x 191 x 1,0211.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 3,414,224.

3,414,224
Factor Arrows
21,707,112
Factor Arrows
2853,556
Factor Arrows
2426,778
Factor Arrows
2213,389
Factor Arrows
1119,399
Factor Arrows
191,021

More Prime Factorization Examples

3,414,2223,414,2233,414,2253,414,226
21 x 32 x 74 x 7913,414,223131 x 52 x 45,523121 x 1,707,1131

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