Q: What is the prime factorization of the number 262,312,128?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 13 x 11,677
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 3, 3, 13, 11,677 }
  • Written in exponential form: 26 x 33 x 131 x 11,6771

Why is the prime factorization of 262,312,128 written as 26 x 33 x 131 x 11,6771?

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 262,312,128

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 262,312,128 by 2

262,312,128 ÷ 2 = 131,156,064 - No remainder! 2 is one of the factors!
131,156,064 ÷ 2 = 65,578,032 - No remainder! 2 is one of the factors!
65,578,032 ÷ 2 = 32,789,016 - No remainder! 2 is one of the factors!
32,789,016 ÷ 2 = 16,394,508 - No remainder! 2 is one of the factors!
16,394,508 ÷ 2 = 8,197,254 - No remainder! 2 is one of the factors!
8,197,254 ÷ 2 = 4,098,627 - No remainder! 2 is one of the factors!
4,098,627 ÷ 2 = 2,049,313.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
4,098,627 ÷ 3 = 1,366,209 - No remainder! 3 is one of the factors!
1,366,209 ÷ 3 = 455,403 - No remainder! 3 is one of the factors!
455,403 ÷ 3 = 151,801 - No remainder! 3 is one of the factors!
151,801 ÷ 3 = 50,600.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
151,801 ÷ 5 = 30,360.2 - This has a remainder. 5 is not a factor.
151,801 ÷ 7 = 21,685.8571 - This has a remainder. 7 is not a factor.
151,801 ÷ 11 = 13,800.0909 - This has a remainder. 11 is not a factor.
151,801 ÷ 13 = 11,677 - No remainder! 13 is one of the factors!
11,677 ÷ 13 = 898.2308 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
11,677 ÷ 17 = 686.8824 - This has a remainder. 17 is not a factor.
11,677 ÷ 19 = 614.5789 - This has a remainder. 19 is not a factor.
11,677 ÷ 23 = 507.6957 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
11,677 ÷ 11,677 = 1 - No remainder! 11,677 is one of the factors!

The orange divisor(s) above are the prime factors of the number 262,312,128. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 13 x 11,677 = 262,312,128. It can also be written in exponential form as 26 x 33 x 131 x 11,6771.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 262,312,128.

262,312,128
Factor Arrows
2131,156,064
Factor Arrows
265,578,032
Factor Arrows
232,789,016
Factor Arrows
216,394,508
Factor Arrows
28,197,254
Factor Arrows
24,098,627
Factor Arrows
31,366,209
Factor Arrows
3455,403
Factor Arrows
3151,801
Factor Arrows
1311,677

More Prime Factorization Examples

262,312,126262,312,127262,312,129262,312,130
21 x 431 x 3,050,141171 x 111 x 711 x 47,9811371 x 7,089,517121 x 51 x 26,231,2131

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