Q: What is the prime factorization of the number 7,522,368?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 7 x 29 x 193
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 7, 29, 193 }
  • Written in exponential form: 26 x 31 x 71 x 291 x 1931

Why is the prime factorization of 7,522,368 written as 26 x 31 x 71 x 291 x 1931?

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 7,522,368

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 7,522,368 by 2

7,522,368 ÷ 2 = 3,761,184 - No remainder! 2 is one of the factors!
3,761,184 ÷ 2 = 1,880,592 - No remainder! 2 is one of the factors!
1,880,592 ÷ 2 = 940,296 - No remainder! 2 is one of the factors!
940,296 ÷ 2 = 470,148 - No remainder! 2 is one of the factors!
470,148 ÷ 2 = 235,074 - No remainder! 2 is one of the factors!
235,074 ÷ 2 = 117,537 - No remainder! 2 is one of the factors!
117,537 ÷ 2 = 58,768.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
117,537 ÷ 3 = 39,179 - No remainder! 3 is one of the factors!
39,179 ÷ 3 = 13,059.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
39,179 ÷ 5 = 7,835.8 - This has a remainder. 5 is not a factor.
39,179 ÷ 7 = 5,597 - No remainder! 7 is one of the factors!
5,597 ÷ 7 = 799.5714 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
5,597 ÷ 11 = 508.8182 - This has a remainder. 11 is not a factor.
5,597 ÷ 13 = 430.5385 - This has a remainder. 13 is not a factor.
5,597 ÷ 17 = 329.2353 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
5,597 ÷ 29 = 193 - No remainder! 29 is one of the factors!
193 ÷ 29 = 6.6552 - There is a remainder. We can't divide by 29 evenly anymore. Let's try the next prime number
193 ÷ 31 = 6.2258 - This has a remainder. 31 is not a factor.
193 ÷ 37 = 5.2162 - This has a remainder. 37 is not a factor.
193 ÷ 41 = 4.7073 - This has a remainder. 41 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
193 ÷ 193 = 1 - No remainder! 193 is one of the factors!

The orange divisor(s) above are the prime factors of the number 7,522,368. 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 7 x 29 x 193 = 7,522,368. It can also be written in exponential form as 26 x 31 x 71 x 291 x 1931.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 7,522,368.

7,522,368
Factor Arrows
23,761,184
Factor Arrows
21,880,592
Factor Arrows
2940,296
Factor Arrows
2470,148
Factor Arrows
2235,074
Factor Arrows
2117,537
Factor Arrows
339,179
Factor Arrows
75,597
Factor Arrows
29193

More Prime Factorization Examples

7,522,3667,522,3677,522,3697,522,370
21 x 191 x 197,9571311 x 1511 x 1,6071891 x 84,521121 x 51 x 2811 x 2,6771

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