Q: What is the prime factorization of the number 103,001,231?

Why is the prime factorization of 103,001,231 written as 241¹ x 257¹ x 1,663¹?

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 103,001,231

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.

Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Let's start by dividing 103,001,231 by 2

103,001,231 ÷ 2 = 51,500,615.5 - This has a remainder. Let's try another prime number.
103,001,231 ÷ 3 = 34,333,743.6667 - This has a remainder. Let's try another prime number.
103,001,231 ÷ 5 = 20,600,246.2 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
103,001,231 ÷ 241 = 427,391 - No remainder! 241 is one of the factors!
427,391 ÷ 241 = 1,773.4066 - There is a remainder. We can't divide by 241 evenly anymore. Let's try the next prime number
427,391 ÷ 251 = 1,702.753 - This has a remainder. 251 is not a factor.
427,391 ÷ 257 = 1,663 - No remainder! 257 is one of the factors!
1,663 ÷ 257 = 6.4708 - There is a remainder. We can't divide by 257 evenly anymore. Let's try the next prime number
1,663 ÷ 263 = 6.3232 - This has a remainder. 263 is not a factor.
1,663 ÷ 269 = 6.1822 - This has a remainder. 269 is not a factor.
1,663 ÷ 271 = 6.1365 - This has a remainder. 271 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,663 ÷ 1,663 = 1 - No remainder! 1,663 is one of the factors!

The orange divisor(s) above are the prime factors of the number 103,001,231. If we put all of it together we have the factors 241 x 257 x 1,663 = 103,001,231. It can also be written in exponential form as 241¹ x 257¹ x 1,663¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 103,001,231.

	103,001,231
241		427,391
	257		1,663

More Prime Factorization Examples

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