Q: What is the prime factorization of the number 7,654,400?

 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 5 x 5 x 13 x 23
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 13, 23 }
  • Written in exponential form: 210 x 52 x 131 x 231

Why is the prime factorization of 7,654,400 written as 210 x 52 x 131 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 7,654,400

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 7,654,400 by 2

7,654,400 ÷ 2 = 3,827,200 - No remainder! 2 is one of the factors!
3,827,200 ÷ 2 = 1,913,600 - No remainder! 2 is one of the factors!
1,913,600 ÷ 2 = 956,800 - No remainder! 2 is one of the factors!
956,800 ÷ 2 = 478,400 - No remainder! 2 is one of the factors!
478,400 ÷ 2 = 239,200 - No remainder! 2 is one of the factors!
239,200 ÷ 2 = 119,600 - No remainder! 2 is one of the factors!
119,600 ÷ 2 = 59,800 - No remainder! 2 is one of the factors!
59,800 ÷ 2 = 29,900 - No remainder! 2 is one of the factors!
29,900 ÷ 2 = 14,950 - No remainder! 2 is one of the factors!
14,950 ÷ 2 = 7,475 - No remainder! 2 is one of the factors!
7,475 ÷ 2 = 3,737.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
7,475 ÷ 3 = 2,491.6667 - This has a remainder. 3 is not a factor.
7,475 ÷ 5 = 1,495 - No remainder! 5 is one of the factors!
1,495 ÷ 5 = 299 - No remainder! 5 is one of the factors!
299 ÷ 5 = 59.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
299 ÷ 7 = 42.7143 - This has a remainder. 7 is not a factor.
299 ÷ 11 = 27.1818 - This has a remainder. 11 is not a factor.
299 ÷ 13 = 23 - No remainder! 13 is one of the factors!
23 ÷ 13 = 1.7692 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
23 ÷ 17 = 1.3529 - This has a remainder. 17 is not a factor.
23 ÷ 19 = 1.2105 - This has a remainder. 19 is not a factor.
23 ÷ 23 = 1 - No remainder! 23 is one of the factors!

The orange divisor(s) above are the prime factors of the number 7,654,400. 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 2 x 2 x 5 x 5 x 13 x 23 = 7,654,400. It can also be written in exponential form as 210 x 52 x 131 x 231.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 7,654,400.

7,654,400
Factor Arrows
23,827,200
Factor Arrows
21,913,600
Factor Arrows
2956,800
Factor Arrows
2478,400
Factor Arrows
2239,200
Factor Arrows
2119,600
Factor Arrows
259,800
Factor Arrows
229,900
Factor Arrows
214,950
Factor Arrows
27,475
Factor Arrows
51,495
Factor Arrows
5299
Factor Arrows
1323

More Prime Factorization Examples

7,654,3987,654,3997,654,4017,654,402
21 x 31 x 1791 x 7,12717,654,399132 x 2411 x 3,529121 x 71 x 611 x 8,9631

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