Q: What is the prime factorization of the number 164,496?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 23 x 149
    • or also written as { 2, 2, 2, 2, 3, 23, 149 }
  • Written in exponential form: 24 x 31 x 231 x 1491

Why is the prime factorization of 164,496 written as 24 x 31 x 231 x 1491?

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 164,496

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 164,496 by 2

164,496 ÷ 2 = 82,248 - No remainder! 2 is one of the factors!
82,248 ÷ 2 = 41,124 - No remainder! 2 is one of the factors!
41,124 ÷ 2 = 20,562 - No remainder! 2 is one of the factors!
20,562 ÷ 2 = 10,281 - No remainder! 2 is one of the factors!
10,281 ÷ 2 = 5,140.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
10,281 ÷ 3 = 3,427 - No remainder! 3 is one of the factors!
3,427 ÷ 3 = 1,142.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
3,427 ÷ 5 = 685.4 - This has a remainder. 5 is not a factor.
3,427 ÷ 7 = 489.5714 - This has a remainder. 7 is not a factor.
3,427 ÷ 11 = 311.5455 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
3,427 ÷ 23 = 149 - No remainder! 23 is one of the factors!
149 ÷ 23 = 6.4783 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
149 ÷ 29 = 5.1379 - This has a remainder. 29 is not a factor.
149 ÷ 31 = 4.8065 - This has a remainder. 31 is not a factor.
149 ÷ 37 = 4.027 - This has a remainder. 37 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
149 ÷ 149 = 1 - No remainder! 149 is one of the factors!

The orange divisor(s) above are the prime factors of the number 164,496. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 23 x 149 = 164,496. It can also be written in exponential form as 24 x 31 x 231 x 1491.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 164,496.

164,496
Factor Arrows
282,248
Factor Arrows
241,124
Factor Arrows
220,562
Factor Arrows
210,281
Factor Arrows
33,427
Factor Arrows
23149

More Prime Factorization Examples

164,494164,495164,497164,498
21 x 111 x 7,477151 x 1671 x 19712711 x 607121 x 2331 x 3531

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