Q: What is the prime factorization of the number 103,000,016?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 7 x 491 x 1,873
    • or also written as { 2, 2, 2, 2, 7, 491, 1,873 }
  • Written in exponential form: 24 x 71 x 4911 x 1,8731

Why is the prime factorization of 103,000,016 written as 24 x 71 x 4911 x 1,8731?

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 103,000,016

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 103,000,016 by 2

103,000,016 ÷ 2 = 51,500,008 - No remainder! 2 is one of the factors!
51,500,008 ÷ 2 = 25,750,004 - No remainder! 2 is one of the factors!
25,750,004 ÷ 2 = 12,875,002 - No remainder! 2 is one of the factors!
12,875,002 ÷ 2 = 6,437,501 - No remainder! 2 is one of the factors!
6,437,501 ÷ 2 = 3,218,750.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
6,437,501 ÷ 3 = 2,145,833.6667 - This has a remainder. 3 is not a factor.
6,437,501 ÷ 5 = 1,287,500.2 - This has a remainder. 5 is not a factor.
6,437,501 ÷ 7 = 919,643 - No remainder! 7 is one of the factors!
919,643 ÷ 7 = 131,377.5714 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
919,643 ÷ 11 = 83,603.9091 - This has a remainder. 11 is not a factor.
919,643 ÷ 13 = 70,741.7692 - This has a remainder. 13 is not a factor.
919,643 ÷ 17 = 54,096.6471 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
919,643 ÷ 491 = 1,873 - No remainder! 491 is one of the factors!
1,873 ÷ 491 = 3.8147 - There is a remainder. We can't divide by 491 evenly anymore. Let's try the next prime number
1,873 ÷ 499 = 3.7535 - This has a remainder. 499 is not a factor.
1,873 ÷ 503 = 3.7237 - This has a remainder. 503 is not a factor.
1,873 ÷ 509 = 3.6798 - This has a remainder. 509 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,873 ÷ 1,873 = 1 - No remainder! 1,873 is one of the factors!

The orange divisor(s) above are the prime factors of the number 103,000,016. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 7 x 491 x 1,873 = 103,000,016. It can also be written in exponential form as 24 x 71 x 4911 x 1,8731.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 103,000,016.

103,000,016
Factor Arrows
251,500,008
Factor Arrows
225,750,004
Factor Arrows
212,875,002
Factor Arrows
26,437,501
Factor Arrows
7919,643
Factor Arrows
4911,873

More Prime Factorization Examples

103,000,014103,000,015103,000,017103,000,018
21 x 32 x 131 x 440,171151 x 20,600,003131 x 591 x 581,921121 x 111 x 3671 x 12,7571

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