Q: What is the prime factorization of the number 211,040,256?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 7 x 7 x 701
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 7, 7, 701 }
  • Written in exponential form: 211 x 31 x 72 x 7011

Why is the prime factorization of 211,040,256 written as 211 x 31 x 72 x 7011?

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 211,040,256

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 211,040,256 by 2

211,040,256 ÷ 2 = 105,520,128 - No remainder! 2 is one of the factors!
105,520,128 ÷ 2 = 52,760,064 - No remainder! 2 is one of the factors!
52,760,064 ÷ 2 = 26,380,032 - No remainder! 2 is one of the factors!
26,380,032 ÷ 2 = 13,190,016 - No remainder! 2 is one of the factors!
13,190,016 ÷ 2 = 6,595,008 - No remainder! 2 is one of the factors!
6,595,008 ÷ 2 = 3,297,504 - No remainder! 2 is one of the factors!
3,297,504 ÷ 2 = 1,648,752 - No remainder! 2 is one of the factors!
1,648,752 ÷ 2 = 824,376 - No remainder! 2 is one of the factors!
824,376 ÷ 2 = 412,188 - No remainder! 2 is one of the factors!
412,188 ÷ 2 = 206,094 - No remainder! 2 is one of the factors!
206,094 ÷ 2 = 103,047 - No remainder! 2 is one of the factors!
103,047 ÷ 2 = 51,523.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
103,047 ÷ 3 = 34,349 - No remainder! 3 is one of the factors!
34,349 ÷ 3 = 11,449.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
34,349 ÷ 5 = 6,869.8 - This has a remainder. 5 is not a factor.
34,349 ÷ 7 = 4,907 - No remainder! 7 is one of the factors!
4,907 ÷ 7 = 701 - No remainder! 7 is one of the factors!
701 ÷ 7 = 100.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
701 ÷ 11 = 63.7273 - This has a remainder. 11 is not a factor.
701 ÷ 13 = 53.9231 - This has a remainder. 13 is not a factor.
701 ÷ 17 = 41.2353 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
701 ÷ 701 = 1 - No remainder! 701 is one of the factors!

The orange divisor(s) above are the prime factors of the number 211,040,256. 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 2 x 2 x 2 x 3 x 7 x 7 x 701 = 211,040,256. It can also be written in exponential form as 211 x 31 x 72 x 7011.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 211,040,256.

211,040,256
Factor Arrows
2105,520,128
Factor Arrows
252,760,064
Factor Arrows
226,380,032
Factor Arrows
213,190,016
Factor Arrows
26,595,008
Factor Arrows
23,297,504
Factor Arrows
21,648,752
Factor Arrows
2824,376
Factor Arrows
2412,188
Factor Arrows
2206,094
Factor Arrows
2103,047
Factor Arrows
334,349
Factor Arrows
74,907
Factor Arrows
7701

More Prime Factorization Examples

211,040,254211,040,255211,040,257211,040,258
21 x 2,2511 x 46,877151 x 42,208,051110,3691 x 20,353121 x 111 x 131 x 191 x 711 x 5471

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