Q: What is the prime factorization of the number 210,414,352?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2,417 x 5,441
    • or also written as { 2, 2, 2, 2, 2,417, 5,441 }
  • Written in exponential form: 24 x 2,4171 x 5,4411

Why is the prime factorization of 210,414,352 written as 24 x 2,4171 x 5,4411?

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 210,414,352

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 210,414,352 by 2

210,414,352 ÷ 2 = 105,207,176 - No remainder! 2 is one of the factors!
105,207,176 ÷ 2 = 52,603,588 - No remainder! 2 is one of the factors!
52,603,588 ÷ 2 = 26,301,794 - No remainder! 2 is one of the factors!
26,301,794 ÷ 2 = 13,150,897 - No remainder! 2 is one of the factors!
13,150,897 ÷ 2 = 6,575,448.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
13,150,897 ÷ 3 = 4,383,632.3333 - This has a remainder. 3 is not a factor.
13,150,897 ÷ 5 = 2,630,179.4 - This has a remainder. 5 is not a factor.
13,150,897 ÷ 7 = 1,878,699.5714 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
13,150,897 ÷ 2,417 = 5,441 - No remainder! 2,417 is one of the factors!
5,441 ÷ 2,417 = 2.2511 - There is a remainder. We can't divide by 2417 evenly anymore. Let's try the next prime number
5,441 ÷ 2,423 = 2.2456 - This has a remainder. 2,423 is not a factor.
5,441 ÷ 2,437 = 2.2327 - This has a remainder. 2,437 is not a factor.
5,441 ÷ 2,441 = 2.229 - This has a remainder. 2,441 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
5,441 ÷ 5,441 = 1 - No remainder! 5,441 is one of the factors!

The orange divisor(s) above are the prime factors of the number 210,414,352. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2,417 x 5,441 = 210,414,352. It can also be written in exponential form as 24 x 2,4171 x 5,4411.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 210,414,352.

210,414,352
Factor Arrows
2105,207,176
Factor Arrows
252,603,588
Factor Arrows
226,301,794
Factor Arrows
213,150,897
Factor Arrows
2,4175,441

More Prime Factorization Examples

210,414,350210,414,351210,414,353210,414,354
21 x 52 x 231 x 182,969131 x 71 x 432 x 5,41917,6691 x 27,437121 x 31 x 1,6991 x 20,6411

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