Q: What is the prime factorization of the number 25,392,640?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 7 x 13 x 109
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 7, 13, 109 }
  • Written in exponential form: 29 x 51 x 71 x 131 x 1091

Why is the prime factorization of 25,392,640 written as 29 x 51 x 71 x 131 x 1091?

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 25,392,640

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 25,392,640 by 2

25,392,640 ÷ 2 = 12,696,320 - No remainder! 2 is one of the factors!
12,696,320 ÷ 2 = 6,348,160 - No remainder! 2 is one of the factors!
6,348,160 ÷ 2 = 3,174,080 - No remainder! 2 is one of the factors!
3,174,080 ÷ 2 = 1,587,040 - No remainder! 2 is one of the factors!
1,587,040 ÷ 2 = 793,520 - No remainder! 2 is one of the factors!
793,520 ÷ 2 = 396,760 - No remainder! 2 is one of the factors!
396,760 ÷ 2 = 198,380 - No remainder! 2 is one of the factors!
198,380 ÷ 2 = 99,190 - No remainder! 2 is one of the factors!
99,190 ÷ 2 = 49,595 - No remainder! 2 is one of the factors!
49,595 ÷ 2 = 24,797.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
49,595 ÷ 3 = 16,531.6667 - This has a remainder. 3 is not a factor.
49,595 ÷ 5 = 9,919 - No remainder! 5 is one of the factors!
9,919 ÷ 5 = 1,983.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
9,919 ÷ 7 = 1,417 - No remainder! 7 is one of the factors!
1,417 ÷ 7 = 202.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
1,417 ÷ 11 = 128.8182 - This has a remainder. 11 is not a factor.
1,417 ÷ 13 = 109 - No remainder! 13 is one of the factors!
109 ÷ 13 = 8.3846 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
109 ÷ 17 = 6.4118 - This has a remainder. 17 is not a factor.
109 ÷ 19 = 5.7368 - This has a remainder. 19 is not a factor.
109 ÷ 23 = 4.7391 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
109 ÷ 109 = 1 - No remainder! 109 is one of the factors!

The orange divisor(s) above are the prime factors of the number 25,392,640. 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 2 x 5 x 7 x 13 x 109 = 25,392,640. It can also be written in exponential form as 29 x 51 x 71 x 131 x 1091.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 25,392,640.

25,392,640
Factor Arrows
212,696,320
Factor Arrows
26,348,160
Factor Arrows
23,174,080
Factor Arrows
21,587,040
Factor Arrows
2793,520
Factor Arrows
2396,760
Factor Arrows
2198,380
Factor Arrows
299,190
Factor Arrows
249,595
Factor Arrows
59,919
Factor Arrows
71,417
Factor Arrows
13109

More Prime Factorization Examples

25,392,63825,392,63925,392,64125,392,642
21 x 4391 x 28,921131 x 8,464,213125,392,641121 x 31 x 111 x 384,7371

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