Q: What is the prime factorization of the number 460,241,408?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 11 x 11 x 17 x 19 x 23
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 11, 11, 17, 19, 23 }
  • Written in exponential form: 29 x 112 x 171 x 191 x 231

Why is the prime factorization of 460,241,408 written as 29 x 112 x 171 x 191 x 231?

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 460,241,408

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 460,241,408 by 2

460,241,408 ÷ 2 = 230,120,704 - No remainder! 2 is one of the factors!
230,120,704 ÷ 2 = 115,060,352 - No remainder! 2 is one of the factors!
115,060,352 ÷ 2 = 57,530,176 - No remainder! 2 is one of the factors!
57,530,176 ÷ 2 = 28,765,088 - No remainder! 2 is one of the factors!
28,765,088 ÷ 2 = 14,382,544 - No remainder! 2 is one of the factors!
14,382,544 ÷ 2 = 7,191,272 - No remainder! 2 is one of the factors!
7,191,272 ÷ 2 = 3,595,636 - No remainder! 2 is one of the factors!
3,595,636 ÷ 2 = 1,797,818 - No remainder! 2 is one of the factors!
1,797,818 ÷ 2 = 898,909 - No remainder! 2 is one of the factors!
898,909 ÷ 2 = 449,454.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
898,909 ÷ 3 = 299,636.3333 - This has a remainder. 3 is not a factor.
898,909 ÷ 5 = 179,781.8 - This has a remainder. 5 is not a factor.
898,909 ÷ 7 = 128,415.5714 - This has a remainder. 7 is not a factor.
898,909 ÷ 11 = 81,719 - No remainder! 11 is one of the factors!
81,719 ÷ 11 = 7,429 - No remainder! 11 is one of the factors!
7,429 ÷ 11 = 675.3636 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
7,429 ÷ 13 = 571.4615 - This has a remainder. 13 is not a factor.
7,429 ÷ 17 = 437 - No remainder! 17 is one of the factors!
437 ÷ 17 = 25.7059 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
437 ÷ 19 = 23 - No remainder! 19 is one of the factors!
23 ÷ 19 = 1.2105 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
23 ÷ 23 = 1 - No remainder! 23 is one of the factors!

The orange divisor(s) above are the prime factors of the number 460,241,408. 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 11 x 11 x 17 x 19 x 23 = 460,241,408. It can also be written in exponential form as 29 x 112 x 171 x 191 x 231.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 460,241,408.

460,241,408
Factor Arrows
2230,120,704
Factor Arrows
2115,060,352
Factor Arrows
257,530,176
Factor Arrows
228,765,088
Factor Arrows
214,382,544
Factor Arrows
27,191,272
Factor Arrows
23,595,636
Factor Arrows
21,797,818
Factor Arrows
2898,909
Factor Arrows
1181,719
Factor Arrows
117,429
Factor Arrows
17437
Factor Arrows
1923

More Prime Factorization Examples

460,241,406460,241,407460,241,409460,241,410
21 x 33 x 8,522,9891311 x 1,0491 x 14,153131 x 371 x 1,2171 x 3,407121 x 51 x 4211 x 109,3211

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