Q: What is the prime factorization of the number 33,984?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 59
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 3, 59 }
  • Written in exponential form: 26 x 32 x 591

Why is the prime factorization of 33,984 written as 26 x 32 x 591?

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 33,984

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 33,984 by 2

33,984 ÷ 2 = 16,992 - No remainder! 2 is one of the factors!
16,992 ÷ 2 = 8,496 - No remainder! 2 is one of the factors!
8,496 ÷ 2 = 4,248 - No remainder! 2 is one of the factors!
4,248 ÷ 2 = 2,124 - No remainder! 2 is one of the factors!
2,124 ÷ 2 = 1,062 - No remainder! 2 is one of the factors!
1,062 ÷ 2 = 531 - No remainder! 2 is one of the factors!
531 ÷ 2 = 265.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
531 ÷ 3 = 177 - No remainder! 3 is one of the factors!
177 ÷ 3 = 59 - No remainder! 3 is one of the factors!
59 ÷ 3 = 19.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
59 ÷ 5 = 11.8 - This has a remainder. 5 is not a factor.
59 ÷ 7 = 8.4286 - This has a remainder. 7 is not a factor.
59 ÷ 11 = 5.3636 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
59 ÷ 59 = 1 - No remainder! 59 is one of the factors!

The orange divisor(s) above are the prime factors of the number 33,984. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 59 = 33,984. It can also be written in exponential form as 26 x 32 x 591.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 33,984.

33,984
Factor Arrows
216,992
Factor Arrows
28,496
Factor Arrows
24,248
Factor Arrows
22,124
Factor Arrows
21,062
Factor Arrows
2531
Factor Arrows
3177
Factor Arrows
359

More Prime Factorization Examples

33,98233,98333,98533,986
21 x 131 x 1,3071171 x 1,999151 x 71 x 971121 x 16,9931

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