Q: What is the prime factorization of the number 125,446,400?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 17 x 1,153
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 17, 1,153 }
  • Written in exponential form: 28 x 52 x 171 x 1,1531

Why is the prime factorization of 125,446,400 written as 28 x 52 x 171 x 1,1531?

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 125,446,400

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 125,446,400 by 2

125,446,400 ÷ 2 = 62,723,200 - No remainder! 2 is one of the factors!
62,723,200 ÷ 2 = 31,361,600 - No remainder! 2 is one of the factors!
31,361,600 ÷ 2 = 15,680,800 - No remainder! 2 is one of the factors!
15,680,800 ÷ 2 = 7,840,400 - No remainder! 2 is one of the factors!
7,840,400 ÷ 2 = 3,920,200 - No remainder! 2 is one of the factors!
3,920,200 ÷ 2 = 1,960,100 - No remainder! 2 is one of the factors!
1,960,100 ÷ 2 = 980,050 - No remainder! 2 is one of the factors!
980,050 ÷ 2 = 490,025 - No remainder! 2 is one of the factors!
490,025 ÷ 2 = 245,012.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
490,025 ÷ 3 = 163,341.6667 - This has a remainder. 3 is not a factor.
490,025 ÷ 5 = 98,005 - No remainder! 5 is one of the factors!
98,005 ÷ 5 = 19,601 - No remainder! 5 is one of the factors!
19,601 ÷ 5 = 3,920.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
19,601 ÷ 7 = 2,800.1429 - This has a remainder. 7 is not a factor.
19,601 ÷ 11 = 1,781.9091 - This has a remainder. 11 is not a factor.
19,601 ÷ 13 = 1,507.7692 - This has a remainder. 13 is not a factor.
19,601 ÷ 17 = 1,153 - No remainder! 17 is one of the factors!
1,153 ÷ 17 = 67.8235 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
1,153 ÷ 19 = 60.6842 - This has a remainder. 19 is not a factor.
1,153 ÷ 23 = 50.1304 - This has a remainder. 23 is not a factor.
1,153 ÷ 29 = 39.7586 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,153 ÷ 1,153 = 1 - No remainder! 1,153 is one of the factors!

The orange divisor(s) above are the prime factors of the number 125,446,400. 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 5 x 5 x 17 x 1,153 = 125,446,400. It can also be written in exponential form as 28 x 52 x 171 x 1,1531.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 125,446,400.

125,446,400
Factor Arrows
262,723,200
Factor Arrows
231,361,600
Factor Arrows
215,680,800
Factor Arrows
27,840,400
Factor Arrows
23,920,200
Factor Arrows
21,960,100
Factor Arrows
2980,050
Factor Arrows
2490,025
Factor Arrows
598,005
Factor Arrows
519,601
Factor Arrows
171,153

More Prime Factorization Examples

125,446,398125,446,399125,446,401125,446,402
21 x 31 x 71 x 111 x 191 x 311 x 4611131 x 3471 x 27,809134 x 1,548,721121 x 291 x 2,162,8691

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