Q: What is the prime factorization of the number 34,503,120?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 3 x 5 x 173 x 277
    • or also written as { 2, 2, 2, 2, 3, 3, 5, 173, 277 }
  • Written in exponential form: 24 x 32 x 51 x 1731 x 2771

Why is the prime factorization of 34,503,120 written as 24 x 32 x 51 x 1731 x 2771?

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 34,503,120

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 34,503,120 by 2

34,503,120 ÷ 2 = 17,251,560 - No remainder! 2 is one of the factors!
17,251,560 ÷ 2 = 8,625,780 - No remainder! 2 is one of the factors!
8,625,780 ÷ 2 = 4,312,890 - No remainder! 2 is one of the factors!
4,312,890 ÷ 2 = 2,156,445 - No remainder! 2 is one of the factors!
2,156,445 ÷ 2 = 1,078,222.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
2,156,445 ÷ 3 = 718,815 - No remainder! 3 is one of the factors!
718,815 ÷ 3 = 239,605 - No remainder! 3 is one of the factors!
239,605 ÷ 3 = 79,868.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
239,605 ÷ 5 = 47,921 - No remainder! 5 is one of the factors!
47,921 ÷ 5 = 9,584.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
47,921 ÷ 7 = 6,845.8571 - This has a remainder. 7 is not a factor.
47,921 ÷ 11 = 4,356.4545 - This has a remainder. 11 is not a factor.
47,921 ÷ 13 = 3,686.2308 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
47,921 ÷ 173 = 277 - No remainder! 173 is one of the factors!
277 ÷ 173 = 1.6012 - There is a remainder. We can't divide by 173 evenly anymore. Let's try the next prime number
277 ÷ 179 = 1.5475 - This has a remainder. 179 is not a factor.
277 ÷ 181 = 1.5304 - This has a remainder. 181 is not a factor.
277 ÷ 191 = 1.4503 - This has a remainder. 191 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
277 ÷ 277 = 1 - No remainder! 277 is one of the factors!

The orange divisor(s) above are the prime factors of the number 34,503,120. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 3 x 5 x 173 x 277 = 34,503,120. It can also be written in exponential form as 24 x 32 x 51 x 1731 x 2771.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 34,503,120.

34,503,120
Factor Arrows
217,251,560
Factor Arrows
28,625,780
Factor Arrows
24,312,890
Factor Arrows
22,156,445
Factor Arrows
3718,815
Factor Arrows
3239,605
Factor Arrows
547,921
Factor Arrows
173277

More Prime Factorization Examples

34,503,11834,503,11934,503,12134,503,122
21 x 131 x 1,327,043171 x 5471 x 9,011134,503,121121 x 17,251,5611

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