Q: What is the prime factorization of the number 3,456,432?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 7 x 127
    • or also written as { 2, 2, 2, 2, 3, 3, 3, 3, 3, 7, 127 }
  • Written in exponential form: 24 x 35 x 71 x 1271

Why is the prime factorization of 3,456,432 written as 24 x 35 x 71 x 1271?

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 3,456,432

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 3,456,432 by 2

3,456,432 ÷ 2 = 1,728,216 - No remainder! 2 is one of the factors!
1,728,216 ÷ 2 = 864,108 - No remainder! 2 is one of the factors!
864,108 ÷ 2 = 432,054 - No remainder! 2 is one of the factors!
432,054 ÷ 2 = 216,027 - No remainder! 2 is one of the factors!
216,027 ÷ 2 = 108,013.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
216,027 ÷ 3 = 72,009 - No remainder! 3 is one of the factors!
72,009 ÷ 3 = 24,003 - No remainder! 3 is one of the factors!
24,003 ÷ 3 = 8,001 - No remainder! 3 is one of the factors!
8,001 ÷ 3 = 2,667 - No remainder! 3 is one of the factors!
2,667 ÷ 3 = 889 - No remainder! 3 is one of the factors!
889 ÷ 3 = 296.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
889 ÷ 5 = 177.8 - This has a remainder. 5 is not a factor.
889 ÷ 7 = 127 - No remainder! 7 is one of the factors!
127 ÷ 7 = 18.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
127 ÷ 11 = 11.5455 - This has a remainder. 11 is not a factor.
127 ÷ 13 = 9.7692 - This has a remainder. 13 is not a factor.
127 ÷ 17 = 7.4706 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
127 ÷ 127 = 1 - No remainder! 127 is one of the factors!

The orange divisor(s) above are the prime factors of the number 3,456,432. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 7 x 127 = 3,456,432. It can also be written in exponential form as 24 x 35 x 71 x 1271.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 3,456,432.

3,456,432
Factor Arrows
21,728,216
Factor Arrows
2864,108
Factor Arrows
2432,054
Factor Arrows
2216,027
Factor Arrows
372,009
Factor Arrows
324,003
Factor Arrows
38,001
Factor Arrows
32,667
Factor Arrows
3889
Factor Arrows
7127

More Prime Factorization Examples

3,456,4303,456,4313,456,4333,456,434
21 x 51 x 345,6431111 x 1991 x 1,57913,456,433121 x 6171 x 2,8011

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