Q: What is the prime factorization of the number 1,103,856?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 13 x 29 x 61
    • or also written as { 2, 2, 2, 2, 3, 13, 29, 61 }
  • Written in exponential form: 24 x 31 x 131 x 291 x 611

Why is the prime factorization of 1,103,856 written as 24 x 31 x 131 x 291 x 611?

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 1,103,856

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 1,103,856 by 2

1,103,856 ÷ 2 = 551,928 - No remainder! 2 is one of the factors!
551,928 ÷ 2 = 275,964 - No remainder! 2 is one of the factors!
275,964 ÷ 2 = 137,982 - No remainder! 2 is one of the factors!
137,982 ÷ 2 = 68,991 - No remainder! 2 is one of the factors!
68,991 ÷ 2 = 34,495.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
68,991 ÷ 3 = 22,997 - No remainder! 3 is one of the factors!
22,997 ÷ 3 = 7,665.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
22,997 ÷ 5 = 4,599.4 - This has a remainder. 5 is not a factor.
22,997 ÷ 7 = 3,285.2857 - This has a remainder. 7 is not a factor.
22,997 ÷ 11 = 2,090.6364 - This has a remainder. 11 is not a factor.
22,997 ÷ 13 = 1,769 - No remainder! 13 is one of the factors!
1,769 ÷ 13 = 136.0769 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
1,769 ÷ 17 = 104.0588 - This has a remainder. 17 is not a factor.
1,769 ÷ 19 = 93.1053 - This has a remainder. 19 is not a factor.
1,769 ÷ 23 = 76.913 - This has a remainder. 23 is not a factor.
1,769 ÷ 29 = 61 - No remainder! 29 is one of the factors!
61 ÷ 29 = 2.1034 - There is a remainder. We can't divide by 29 evenly anymore. Let's try the next prime number
61 ÷ 31 = 1.9677 - This has a remainder. 31 is not a factor.
61 ÷ 37 = 1.6486 - This has a remainder. 37 is not a factor.
61 ÷ 41 = 1.4878 - This has a remainder. 41 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
61 ÷ 61 = 1 - No remainder! 61 is one of the factors!

The orange divisor(s) above are the prime factors of the number 1,103,856. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 13 x 29 x 61 = 1,103,856. It can also be written in exponential form as 24 x 31 x 131 x 291 x 611.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 1,103,856.

1,103,856
Factor Arrows
2551,928
Factor Arrows
2275,964
Factor Arrows
2137,982
Factor Arrows
268,991
Factor Arrows
322,997
Factor Arrows
131,769
Factor Arrows
2961

More Prime Factorization Examples

1,103,8541,103,8551,103,8571,103,858
21 x 551,927151 x 220,77111,103,857121 x 71 x 371 x 2,1311

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