Q: What is the prime factorization of the number 3,825,585?

 A:
  • The prime factors are: 3 x 3 x 5 x 151 x 563
    • or also written as { 3, 3, 5, 151, 563 }
  • Written in exponential form: 32 x 51 x 1511 x 5631

Why is the prime factorization of 3,825,585 written as 32 x 51 x 1511 x 5631?

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 3,825,585

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 3,825,585 by 2

3,825,585 ÷ 2 = 1,912,792.5 - This has a remainder. Let's try another prime number.
3,825,585 ÷ 3 = 1,275,195 - No remainder! 3 is one of the factors!
1,275,195 ÷ 3 = 425,065 - No remainder! 3 is one of the factors!
425,065 ÷ 3 = 141,688.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
425,065 ÷ 5 = 85,013 - No remainder! 5 is one of the factors!
85,013 ÷ 5 = 17,002.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
85,013 ÷ 7 = 12,144.7143 - This has a remainder. 7 is not a factor.
85,013 ÷ 11 = 7,728.4545 - This has a remainder. 11 is not a factor.
85,013 ÷ 13 = 6,539.4615 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
85,013 ÷ 151 = 563 - No remainder! 151 is one of the factors!
563 ÷ 151 = 3.7285 - There is a remainder. We can't divide by 151 evenly anymore. Let's try the next prime number
563 ÷ 157 = 3.586 - This has a remainder. 157 is not a factor.
563 ÷ 163 = 3.454 - This has a remainder. 163 is not a factor.
563 ÷ 167 = 3.3713 - This has a remainder. 167 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
563 ÷ 563 = 1 - No remainder! 563 is one of the factors!

The orange divisor(s) above are the prime factors of the number 3,825,585. If we put all of it together we have the factors 3 x 3 x 5 x 151 x 563 = 3,825,585. It can also be written in exponential form as 32 x 51 x 1511 x 5631.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 3,825,585.

3,825,585
Factor Arrows
31,275,195
Factor Arrows
3425,065
Factor Arrows
585,013
Factor Arrows
151563

More Prime Factorization Examples

3,825,5833,825,5843,825,5863,825,587
971 x 39,439124 x 71 x 34,157121 x 311 x 61,7031411 x 93,3071

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