Q: What is the prime factorization of the number 10,754,128?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 7 x 7 x 11 x 29 x 43
    • or also written as { 2, 2, 2, 2, 7, 7, 11, 29, 43 }
  • Written in exponential form: 24 x 72 x 111 x 291 x 431

Why is the prime factorization of 10,754,128 written as 24 x 72 x 111 x 291 x 431?

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 10,754,128

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 10,754,128 by 2

10,754,128 ÷ 2 = 5,377,064 - No remainder! 2 is one of the factors!
5,377,064 ÷ 2 = 2,688,532 - No remainder! 2 is one of the factors!
2,688,532 ÷ 2 = 1,344,266 - No remainder! 2 is one of the factors!
1,344,266 ÷ 2 = 672,133 - No remainder! 2 is one of the factors!
672,133 ÷ 2 = 336,066.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
672,133 ÷ 3 = 224,044.3333 - This has a remainder. 3 is not a factor.
672,133 ÷ 5 = 134,426.6 - This has a remainder. 5 is not a factor.
672,133 ÷ 7 = 96,019 - No remainder! 7 is one of the factors!
96,019 ÷ 7 = 13,717 - No remainder! 7 is one of the factors!
13,717 ÷ 7 = 1,959.5714 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
13,717 ÷ 11 = 1,247 - No remainder! 11 is one of the factors!
1,247 ÷ 11 = 113.3636 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
1,247 ÷ 13 = 95.9231 - This has a remainder. 13 is not a factor.
1,247 ÷ 17 = 73.3529 - This has a remainder. 17 is not a factor.
1,247 ÷ 19 = 65.6316 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,247 ÷ 29 = 43 - No remainder! 29 is one of the factors!
43 ÷ 29 = 1.4828 - There is a remainder. We can't divide by 29 evenly anymore. Let's try the next prime number
43 ÷ 31 = 1.3871 - This has a remainder. 31 is not a factor.
43 ÷ 37 = 1.1622 - This has a remainder. 37 is not a factor.
43 ÷ 41 = 1.0488 - This has a remainder. 41 is not a factor.
43 ÷ 43 = 1 - No remainder! 43 is one of the factors!

The orange divisor(s) above are the prime factors of the number 10,754,128. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 7 x 7 x 11 x 29 x 43 = 10,754,128. It can also be written in exponential form as 24 x 72 x 111 x 291 x 431.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 10,754,128.

10,754,128
Factor Arrows
25,377,064
Factor Arrows
22,688,532
Factor Arrows
21,344,266
Factor Arrows
2672,133
Factor Arrows
796,019
Factor Arrows
713,717
Factor Arrows
111,247
Factor Arrows
2943

More Prime Factorization Examples

10,754,12610,754,12710,754,12910,754,130
21 x 5771 x 9,319134 x 1031 x 1,289110,754,129121 x 31 x 51 x 358,4711

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