Q: What is the prime factorization of the number 2,191,104?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 317
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 317 }
  • Written in exponential form: 28 x 33 x 3171

Why is the prime factorization of 2,191,104 written as 28 x 33 x 3171?

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 2,191,104

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 2,191,104 by 2

2,191,104 ÷ 2 = 1,095,552 - No remainder! 2 is one of the factors!
1,095,552 ÷ 2 = 547,776 - No remainder! 2 is one of the factors!
547,776 ÷ 2 = 273,888 - No remainder! 2 is one of the factors!
273,888 ÷ 2 = 136,944 - No remainder! 2 is one of the factors!
136,944 ÷ 2 = 68,472 - No remainder! 2 is one of the factors!
68,472 ÷ 2 = 34,236 - No remainder! 2 is one of the factors!
34,236 ÷ 2 = 17,118 - No remainder! 2 is one of the factors!
17,118 ÷ 2 = 8,559 - No remainder! 2 is one of the factors!
8,559 ÷ 2 = 4,279.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
8,559 ÷ 3 = 2,853 - No remainder! 3 is one of the factors!
2,853 ÷ 3 = 951 - No remainder! 3 is one of the factors!
951 ÷ 3 = 317 - No remainder! 3 is one of the factors!
317 ÷ 3 = 105.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
317 ÷ 5 = 63.4 - This has a remainder. 5 is not a factor.
317 ÷ 7 = 45.2857 - This has a remainder. 7 is not a factor.
317 ÷ 11 = 28.8182 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
317 ÷ 317 = 1 - No remainder! 317 is one of the factors!

The orange divisor(s) above are the prime factors of the number 2,191,104. 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 3 x 3 x 3 x 317 = 2,191,104. It can also be written in exponential form as 28 x 33 x 3171.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 2,191,104.

2,191,104
Factor Arrows
21,095,552
Factor Arrows
2547,776
Factor Arrows
2273,888
Factor Arrows
2136,944
Factor Arrows
268,472
Factor Arrows
234,236
Factor Arrows
217,118
Factor Arrows
28,559
Factor Arrows
32,853
Factor Arrows
3951
Factor Arrows
3317

More Prime Factorization Examples

2,191,1022,191,1032,191,1052,191,106
21 x 1,095,5511371 x 59,219151 x 71 x 62,603121 x 1311 x 8,3631

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