Q: What is the prime factorization of the number 3,733,632?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7 x 463
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 3, 3, 7, 463 }
  • Written in exponential form: 27 x 32 x 71 x 4631

Why is the prime factorization of 3,733,632 written as 27 x 32 x 71 x 4631?

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 3,733,632

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 3,733,632 by 2

3,733,632 ÷ 2 = 1,866,816 - No remainder! 2 is one of the factors!
1,866,816 ÷ 2 = 933,408 - No remainder! 2 is one of the factors!
933,408 ÷ 2 = 466,704 - No remainder! 2 is one of the factors!
466,704 ÷ 2 = 233,352 - No remainder! 2 is one of the factors!
233,352 ÷ 2 = 116,676 - No remainder! 2 is one of the factors!
116,676 ÷ 2 = 58,338 - No remainder! 2 is one of the factors!
58,338 ÷ 2 = 29,169 - No remainder! 2 is one of the factors!
29,169 ÷ 2 = 14,584.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
29,169 ÷ 3 = 9,723 - No remainder! 3 is one of the factors!
9,723 ÷ 3 = 3,241 - No remainder! 3 is one of the factors!
3,241 ÷ 3 = 1,080.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
3,241 ÷ 5 = 648.2 - This has a remainder. 5 is not a factor.
3,241 ÷ 7 = 463 - No remainder! 7 is one of the factors!
463 ÷ 7 = 66.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
463 ÷ 11 = 42.0909 - This has a remainder. 11 is not a factor.
463 ÷ 13 = 35.6154 - This has a remainder. 13 is not a factor.
463 ÷ 17 = 27.2353 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
463 ÷ 463 = 1 - No remainder! 463 is one of the factors!

The orange divisor(s) above are the prime factors of the number 3,733,632. 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 3 x 3 x 7 x 463 = 3,733,632. It can also be written in exponential form as 27 x 32 x 71 x 4631.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 3,733,632.

3,733,632
Factor Arrows
21,866,816
Factor Arrows
2933,408
Factor Arrows
2466,704
Factor Arrows
2233,352
Factor Arrows
2116,676
Factor Arrows
258,338
Factor Arrows
229,169
Factor Arrows
39,723
Factor Arrows
33,241
Factor Arrows
7463

More Prime Factorization Examples

3,733,6303,733,6313,733,6333,733,634
21 x 51 x 373,3631111 x 1311 x 2,5911191 x 371 x 471 x 113121 x 291 x 64,3731

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