Q: What is the prime factorization of the number 103,324,400?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 5 x 5 x 97 x 2,663
    • or also written as { 2, 2, 2, 2, 5, 5, 97, 2,663 }
  • Written in exponential form: 24 x 52 x 971 x 2,6631

Why is the prime factorization of 103,324,400 written as 24 x 52 x 971 x 2,6631?

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 103,324,400

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 103,324,400 by 2

103,324,400 ÷ 2 = 51,662,200 - No remainder! 2 is one of the factors!
51,662,200 ÷ 2 = 25,831,100 - No remainder! 2 is one of the factors!
25,831,100 ÷ 2 = 12,915,550 - No remainder! 2 is one of the factors!
12,915,550 ÷ 2 = 6,457,775 - No remainder! 2 is one of the factors!
6,457,775 ÷ 2 = 3,228,887.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
6,457,775 ÷ 3 = 2,152,591.6667 - This has a remainder. 3 is not a factor.
6,457,775 ÷ 5 = 1,291,555 - No remainder! 5 is one of the factors!
1,291,555 ÷ 5 = 258,311 - No remainder! 5 is one of the factors!
258,311 ÷ 5 = 51,662.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
258,311 ÷ 7 = 36,901.5714 - This has a remainder. 7 is not a factor.
258,311 ÷ 11 = 23,482.8182 - This has a remainder. 11 is not a factor.
258,311 ÷ 13 = 19,870.0769 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
258,311 ÷ 97 = 2,663 - No remainder! 97 is one of the factors!
2,663 ÷ 97 = 27.4536 - There is a remainder. We can't divide by 97 evenly anymore. Let's try the next prime number
2,663 ÷ 101 = 26.3663 - This has a remainder. 101 is not a factor.
2,663 ÷ 103 = 25.8544 - This has a remainder. 103 is not a factor.
2,663 ÷ 107 = 24.8879 - This has a remainder. 107 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,663 ÷ 2,663 = 1 - No remainder! 2,663 is one of the factors!

The orange divisor(s) above are the prime factors of the number 103,324,400. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 5 x 5 x 97 x 2,663 = 103,324,400. It can also be written in exponential form as 24 x 52 x 971 x 2,6631.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 103,324,400.

103,324,400
Factor Arrows
251,662,200
Factor Arrows
225,831,100
Factor Arrows
212,915,550
Factor Arrows
26,457,775
Factor Arrows
51,291,555
Factor Arrows
5258,311
Factor Arrows
972,663

More Prime Factorization Examples

103,324,398103,324,399103,324,401103,324,402
21 x 31 x 1,3611 x 12,6531431 x 591 x 1391 x 293132 x 531 x 2511 x 863121 x 171 x 1,0491 x 2,8971

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