Q: What is the prime factorization of the number 38,119,680?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 1,103
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 5, 1,103 }
  • Written in exponential form: 28 x 33 x 51 x 1,1031

Why is the prime factorization of 38,119,680 written as 28 x 33 x 51 x 1,1031?

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 38,119,680

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 38,119,680 by 2

38,119,680 ÷ 2 = 19,059,840 - No remainder! 2 is one of the factors!
19,059,840 ÷ 2 = 9,529,920 - No remainder! 2 is one of the factors!
9,529,920 ÷ 2 = 4,764,960 - No remainder! 2 is one of the factors!
4,764,960 ÷ 2 = 2,382,480 - No remainder! 2 is one of the factors!
2,382,480 ÷ 2 = 1,191,240 - No remainder! 2 is one of the factors!
1,191,240 ÷ 2 = 595,620 - No remainder! 2 is one of the factors!
595,620 ÷ 2 = 297,810 - No remainder! 2 is one of the factors!
297,810 ÷ 2 = 148,905 - No remainder! 2 is one of the factors!
148,905 ÷ 2 = 74,452.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
148,905 ÷ 3 = 49,635 - No remainder! 3 is one of the factors!
49,635 ÷ 3 = 16,545 - No remainder! 3 is one of the factors!
16,545 ÷ 3 = 5,515 - No remainder! 3 is one of the factors!
5,515 ÷ 3 = 1,838.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
5,515 ÷ 5 = 1,103 - No remainder! 5 is one of the factors!
1,103 ÷ 5 = 220.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
1,103 ÷ 7 = 157.5714 - This has a remainder. 7 is not a factor.
1,103 ÷ 11 = 100.2727 - This has a remainder. 11 is not a factor.
1,103 ÷ 13 = 84.8462 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,103 ÷ 1,103 = 1 - No remainder! 1,103 is one of the factors!

The orange divisor(s) above are the prime factors of the number 38,119,680. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 1,103 = 38,119,680. It can also be written in exponential form as 28 x 33 x 51 x 1,1031.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 38,119,680.

38,119,680
Factor Arrows
219,059,840
Factor Arrows
29,529,920
Factor Arrows
24,764,960
Factor Arrows
22,382,480
Factor Arrows
21,191,240
Factor Arrows
2595,620
Factor Arrows
2297,810
Factor Arrows
2148,905
Factor Arrows
349,635
Factor Arrows
316,545
Factor Arrows
35,515
Factor Arrows
51,103

More Prime Factorization Examples

38,119,67838,119,67938,119,68138,119,682
21 x 172 x 65,9511131 x 471 x 891 x 7011191 x 2,006,299121 x 1031 x 2111 x 8771

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