Q: What is the prime factorization of the number 4,513,408?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 37 x 953
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 37, 953 }
  • Written in exponential form: 27 x 371 x 9531

Why is the prime factorization of 4,513,408 written as 27 x 371 x 9531?

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 4,513,408

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 4,513,408 by 2

4,513,408 ÷ 2 = 2,256,704 - No remainder! 2 is one of the factors!
2,256,704 ÷ 2 = 1,128,352 - No remainder! 2 is one of the factors!
1,128,352 ÷ 2 = 564,176 - No remainder! 2 is one of the factors!
564,176 ÷ 2 = 282,088 - No remainder! 2 is one of the factors!
282,088 ÷ 2 = 141,044 - No remainder! 2 is one of the factors!
141,044 ÷ 2 = 70,522 - No remainder! 2 is one of the factors!
70,522 ÷ 2 = 35,261 - No remainder! 2 is one of the factors!
35,261 ÷ 2 = 17,630.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
35,261 ÷ 3 = 11,753.6667 - This has a remainder. 3 is not a factor.
35,261 ÷ 5 = 7,052.2 - This has a remainder. 5 is not a factor.
35,261 ÷ 7 = 5,037.2857 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
35,261 ÷ 37 = 953 - No remainder! 37 is one of the factors!
953 ÷ 37 = 25.7568 - There is a remainder. We can't divide by 37 evenly anymore. Let's try the next prime number
953 ÷ 41 = 23.2439 - This has a remainder. 41 is not a factor.
953 ÷ 43 = 22.1628 - This has a remainder. 43 is not a factor.
953 ÷ 47 = 20.2766 - This has a remainder. 47 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
953 ÷ 953 = 1 - No remainder! 953 is one of the factors!

The orange divisor(s) above are the prime factors of the number 4,513,408. 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 37 x 953 = 4,513,408. It can also be written in exponential form as 27 x 371 x 9531.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 4,513,408.

4,513,408
Factor Arrows
22,256,704
Factor Arrows
21,128,352
Factor Arrows
2564,176
Factor Arrows
2282,088
Factor Arrows
2141,044
Factor Arrows
270,522
Factor Arrows
235,261
Factor Arrows
37953

More Prime Factorization Examples

4,513,4064,513,4074,513,4094,513,410
21 x 2,256,703131 x 1,504,4691432 x 2,441121 x 32 x 51 x 111 x 471 x 971

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