Q: What is the prime factorization of the number 4,243,456?

 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 2 x 2 x 2 x 2 x 7 x 37
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 37 }
  • Written in exponential form: 214 x 71 x 371

Why is the prime factorization of 4,243,456 written as 214 x 71 x 371?

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 4,243,456

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 4,243,456 by 2

4,243,456 ÷ 2 = 2,121,728 - No remainder! 2 is one of the factors!
2,121,728 ÷ 2 = 1,060,864 - No remainder! 2 is one of the factors!
1,060,864 ÷ 2 = 530,432 - No remainder! 2 is one of the factors!
530,432 ÷ 2 = 265,216 - No remainder! 2 is one of the factors!
265,216 ÷ 2 = 132,608 - No remainder! 2 is one of the factors!
132,608 ÷ 2 = 66,304 - No remainder! 2 is one of the factors!
66,304 ÷ 2 = 33,152 - No remainder! 2 is one of the factors!
33,152 ÷ 2 = 16,576 - No remainder! 2 is one of the factors!
16,576 ÷ 2 = 8,288 - No remainder! 2 is one of the factors!
8,288 ÷ 2 = 4,144 - No remainder! 2 is one of the factors!
4,144 ÷ 2 = 2,072 - No remainder! 2 is one of the factors!
2,072 ÷ 2 = 1,036 - No remainder! 2 is one of the factors!
1,036 ÷ 2 = 518 - No remainder! 2 is one of the factors!
518 ÷ 2 = 259 - No remainder! 2 is one of the factors!
259 ÷ 2 = 129.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
259 ÷ 3 = 86.3333 - This has a remainder. 3 is not a factor.
259 ÷ 5 = 51.8 - This has a remainder. 5 is not a factor.
259 ÷ 7 = 37 - No remainder! 7 is one of the factors!
37 ÷ 7 = 5.2857 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
37 ÷ 11 = 3.3636 - This has a remainder. 11 is not a factor.
37 ÷ 13 = 2.8462 - This has a remainder. 13 is not a factor.
37 ÷ 17 = 2.1765 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
37 ÷ 37 = 1 - No remainder! 37 is one of the factors!

The orange divisor(s) above are the prime factors of the number 4,243,456. 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 2 x 2 x 2 x 2 x 7 x 37 = 4,243,456. It can also be written in exponential form as 214 x 71 x 371.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 4,243,456.

4,243,456
Factor Arrows
22,121,728
Factor Arrows
21,060,864
Factor Arrows
2530,432
Factor Arrows
2265,216
Factor Arrows
2132,608
Factor Arrows
266,304
Factor Arrows
233,152
Factor Arrows
216,576
Factor Arrows
28,288
Factor Arrows
24,144
Factor Arrows
22,072
Factor Arrows
21,036
Factor Arrows
2518
Factor Arrows
2259
Factor Arrows
737

More Prime Factorization Examples

4,243,4544,243,4554,243,4574,243,458
21 x 231 x 291 x 3,181133 x 51 x 171 x 432591 x 711 x 1,013121 x 31 x 831 x 8,5211

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