Q: What is the prime factorization of the number 2,545,272?

 A:
  • The prime factors are: 2 x 2 x 2 x 3 x 3 x 23 x 29 x 53
    • or also written as { 2, 2, 2, 3, 3, 23, 29, 53 }
  • Written in exponential form: 23 x 32 x 231 x 291 x 531

Why is the prime factorization of 2,545,272 written as 23 x 32 x 231 x 291 x 531?

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 2,545,272

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 2,545,272 by 2

2,545,272 ÷ 2 = 1,272,636 - No remainder! 2 is one of the factors!
1,272,636 ÷ 2 = 636,318 - No remainder! 2 is one of the factors!
636,318 ÷ 2 = 318,159 - No remainder! 2 is one of the factors!
318,159 ÷ 2 = 159,079.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
318,159 ÷ 3 = 106,053 - No remainder! 3 is one of the factors!
106,053 ÷ 3 = 35,351 - No remainder! 3 is one of the factors!
35,351 ÷ 3 = 11,783.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
35,351 ÷ 5 = 7,070.2 - This has a remainder. 5 is not a factor.
35,351 ÷ 7 = 5,050.1429 - This has a remainder. 7 is not a factor.
35,351 ÷ 11 = 3,213.7273 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
35,351 ÷ 23 = 1,537 - No remainder! 23 is one of the factors!
1,537 ÷ 23 = 66.8261 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
1,537 ÷ 29 = 53 - No remainder! 29 is one of the factors!
53 ÷ 29 = 1.8276 - There is a remainder. We can't divide by 29 evenly anymore. Let's try the next prime number
53 ÷ 31 = 1.7097 - This has a remainder. 31 is not a factor.
53 ÷ 37 = 1.4324 - This has a remainder. 37 is not a factor.
53 ÷ 41 = 1.2927 - This has a remainder. 41 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
53 ÷ 53 = 1 - No remainder! 53 is one of the factors!

The orange divisor(s) above are the prime factors of the number 2,545,272. If we put all of it together we have the factors 2 x 2 x 2 x 3 x 3 x 23 x 29 x 53 = 2,545,272. It can also be written in exponential form as 23 x 32 x 231 x 291 x 531.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 2,545,272.

2,545,272
Factor Arrows
21,272,636
Factor Arrows
2636,318
Factor Arrows
2318,159
Factor Arrows
3106,053
Factor Arrows
335,351
Factor Arrows
231,537
Factor Arrows
2953

More Prime Factorization Examples

2,545,2702,545,2712,545,2732,545,274
21 x 51 x 71 x 131 x 2,79712,545,27111,4471 x 1,759121 x 171 x 74,8611

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