Q: What is the prime factorization of the number 264,960?

 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 5 x 23
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 5, 23 }
  • Written in exponential form: 28 x 32 x 51 x 231

Why is the prime factorization of 264,960 written as 28 x 32 x 51 x 231?

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 264,960

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 264,960 by 2

264,960 ÷ 2 = 132,480 - No remainder! 2 is one of the factors!
132,480 ÷ 2 = 66,240 - No remainder! 2 is one of the factors!
66,240 ÷ 2 = 33,120 - No remainder! 2 is one of the factors!
33,120 ÷ 2 = 16,560 - No remainder! 2 is one of the factors!
16,560 ÷ 2 = 8,280 - No remainder! 2 is one of the factors!
8,280 ÷ 2 = 4,140 - No remainder! 2 is one of the factors!
4,140 ÷ 2 = 2,070 - No remainder! 2 is one of the factors!
2,070 ÷ 2 = 1,035 - No remainder! 2 is one of the factors!
1,035 ÷ 2 = 517.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,035 ÷ 3 = 345 - No remainder! 3 is one of the factors!
345 ÷ 3 = 115 - No remainder! 3 is one of the factors!
115 ÷ 3 = 38.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
115 ÷ 5 = 23 - No remainder! 5 is one of the factors!
23 ÷ 5 = 4.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
23 ÷ 7 = 3.2857 - This has a remainder. 7 is not a factor.
23 ÷ 11 = 2.0909 - This has a remainder. 11 is not a factor.
23 ÷ 13 = 1.7692 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
23 ÷ 23 = 1 - No remainder! 23 is one of the factors!

The orange divisor(s) above are the prime factors of the number 264,960. 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 5 x 23 = 264,960. It can also be written in exponential form as 28 x 32 x 51 x 231.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 264,960.

264,960
Factor Arrows
2132,480
Factor Arrows
266,240
Factor Arrows
233,120
Factor Arrows
216,560
Factor Arrows
28,280
Factor Arrows
24,140
Factor Arrows
22,070
Factor Arrows
21,035
Factor Arrows
3345
Factor Arrows
3115
Factor Arrows
523

More Prime Factorization Examples

264,958264,959264,961264,962
21 x 1371 x 9671264,9591264,961121 x 171 x 7,7931

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