Q: What is the prime factorization of the number 34,456,224?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 3 x 11 x 67 x 487
    • or also written as { 2, 2, 2, 2, 2, 3, 11, 67, 487 }
  • Written in exponential form: 25 x 31 x 111 x 671 x 4871

Why is the prime factorization of 34,456,224 written as 25 x 31 x 111 x 671 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 34,456,224

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 34,456,224 by 2

34,456,224 ÷ 2 = 17,228,112 - No remainder! 2 is one of the factors!
17,228,112 ÷ 2 = 8,614,056 - No remainder! 2 is one of the factors!
8,614,056 ÷ 2 = 4,307,028 - No remainder! 2 is one of the factors!
4,307,028 ÷ 2 = 2,153,514 - No remainder! 2 is one of the factors!
2,153,514 ÷ 2 = 1,076,757 - No remainder! 2 is one of the factors!
1,076,757 ÷ 2 = 538,378.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,076,757 ÷ 3 = 358,919 - No remainder! 3 is one of the factors!
358,919 ÷ 3 = 119,639.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
358,919 ÷ 5 = 71,783.8 - This has a remainder. 5 is not a factor.
358,919 ÷ 7 = 51,274.1429 - This has a remainder. 7 is not a factor.
358,919 ÷ 11 = 32,629 - No remainder! 11 is one of the factors!
32,629 ÷ 11 = 2,966.2727 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
32,629 ÷ 13 = 2,509.9231 - This has a remainder. 13 is not a factor.
32,629 ÷ 17 = 1,919.3529 - This has a remainder. 17 is not a factor.
32,629 ÷ 19 = 1,717.3158 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
32,629 ÷ 67 = 487 - No remainder! 67 is one of the factors!
487 ÷ 67 = 7.2687 - There is a remainder. We can't divide by 67 evenly anymore. Let's try the next prime number
487 ÷ 71 = 6.8592 - This has a remainder. 71 is not a factor.
487 ÷ 73 = 6.6712 - This has a remainder. 73 is not a factor.
487 ÷ 79 = 6.1646 - This has a remainder. 79 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 34,456,224. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 3 x 11 x 67 x 487 = 34,456,224. It can also be written in exponential form as 25 x 31 x 111 x 671 x 4871.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 34,456,224.

34,456,224
Factor Arrows
217,228,112
Factor Arrows
28,614,056
Factor Arrows
24,307,028
Factor Arrows
22,153,514
Factor Arrows
21,076,757
Factor Arrows
3358,919
Factor Arrows
1132,629
Factor Arrows
67487

More Prime Factorization Examples

34,456,22234,456,22334,456,22534,456,226
21 x 17,228,111134,456,223152 x 1,378,249121 x 71 x 1,1931 x 2,0631

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