Q: What is the prime factorization of the number 334,503?

 A:
  • The prime factors are: 3 x 3 x 3 x 13 x 953
    • or also written as { 3, 3, 3, 13, 953 }
  • Written in exponential form: 33 x 131 x 9531

Why is the prime factorization of 334,503 written as 33 x 131 x 9531?

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 334,503

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 334,503 by 2

334,503 ÷ 2 = 167,251.5 - This has a remainder. Let's try another prime number.
334,503 ÷ 3 = 111,501 - No remainder! 3 is one of the factors!
111,501 ÷ 3 = 37,167 - No remainder! 3 is one of the factors!
37,167 ÷ 3 = 12,389 - No remainder! 3 is one of the factors!
12,389 ÷ 3 = 4,129.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
12,389 ÷ 5 = 2,477.8 - This has a remainder. 5 is not a factor.
12,389 ÷ 7 = 1,769.8571 - This has a remainder. 7 is not a factor.
12,389 ÷ 11 = 1,126.2727 - This has a remainder. 11 is not a factor.
12,389 ÷ 13 = 953 - No remainder! 13 is one of the factors!
953 ÷ 13 = 73.3077 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
953 ÷ 17 = 56.0588 - This has a remainder. 17 is not a factor.
953 ÷ 19 = 50.1579 - This has a remainder. 19 is not a factor.
953 ÷ 23 = 41.4348 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
953 ÷ 953 = 1 - No remainder! 953 is one of the factors!

The orange divisor(s) above are the prime factors of the number 334,503. If we put all of it together we have the factors 3 x 3 x 3 x 13 x 953 = 334,503. It can also be written in exponential form as 33 x 131 x 9531.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 334,503.

334,503
Factor Arrows
3111,501
Factor Arrows
337,167
Factor Arrows
312,389
Factor Arrows
13953

More Prime Factorization Examples

334,501334,502334,504334,505
1671 x 2,003121 x 71 x 23,893123 x 41,813151 x 1491 x 4491

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