Q: What is the prime factorization of the number 4,039,680?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 263
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 263 }
  • Written in exponential form: 210 x 31 x 51 x 2631

Why is the prime factorization of 4,039,680 written as 210 x 31 x 51 x 2631?

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 4,039,680

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 4,039,680 by 2

4,039,680 ÷ 2 = 2,019,840 - No remainder! 2 is one of the factors!
2,019,840 ÷ 2 = 1,009,920 - No remainder! 2 is one of the factors!
1,009,920 ÷ 2 = 504,960 - No remainder! 2 is one of the factors!
504,960 ÷ 2 = 252,480 - No remainder! 2 is one of the factors!
252,480 ÷ 2 = 126,240 - No remainder! 2 is one of the factors!
126,240 ÷ 2 = 63,120 - No remainder! 2 is one of the factors!
63,120 ÷ 2 = 31,560 - No remainder! 2 is one of the factors!
31,560 ÷ 2 = 15,780 - No remainder! 2 is one of the factors!
15,780 ÷ 2 = 7,890 - No remainder! 2 is one of the factors!
7,890 ÷ 2 = 3,945 - No remainder! 2 is one of the factors!
3,945 ÷ 2 = 1,972.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
3,945 ÷ 3 = 1,315 - No remainder! 3 is one of the factors!
1,315 ÷ 3 = 438.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
1,315 ÷ 5 = 263 - No remainder! 5 is one of the factors!
263 ÷ 5 = 52.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
263 ÷ 7 = 37.5714 - This has a remainder. 7 is not a factor.
263 ÷ 11 = 23.9091 - This has a remainder. 11 is not a factor.
263 ÷ 13 = 20.2308 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
263 ÷ 263 = 1 - No remainder! 263 is one of the factors!

The orange divisor(s) above are the prime factors of the number 4,039,680. 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 2 x 3 x 5 x 263 = 4,039,680. It can also be written in exponential form as 210 x 31 x 51 x 2631.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 4,039,680.

4,039,680
Factor Arrows
22,019,840
Factor Arrows
21,009,920
Factor Arrows
2504,960
Factor Arrows
2252,480
Factor Arrows
2126,240
Factor Arrows
263,120
Factor Arrows
231,560
Factor Arrows
215,780
Factor Arrows
27,890
Factor Arrows
23,945
Factor Arrows
31,315
Factor Arrows
5263

More Prime Factorization Examples

4,039,6784,039,6794,039,6814,039,682
21 x 431 x 1071 x 439171 x 577,09714,039,681121 x 1971 x 10,2531

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