Q: What is the prime factorization of the number 110,226,047?

Why is the prime factorization of 110,226,047 written as 97¹ x 101¹ x 11,251¹?

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 110,226,047

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 110,226,047 by 2

110,226,047 ÷ 2 = 55,113,023.5 - This has a remainder. Let's try another prime number.
110,226,047 ÷ 3 = 36,742,015.6667 - This has a remainder. Let's try another prime number.
110,226,047 ÷ 5 = 22,045,209.4 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
110,226,047 ÷ 97 = 1,136,351 - No remainder! 97 is one of the factors!
1,136,351 ÷ 97 = 11,714.9588 - There is a remainder. We can't divide by 97 evenly anymore. Let's try the next prime number
1,136,351 ÷ 101 = 11,251 - No remainder! 101 is one of the factors!
11,251 ÷ 101 = 111.396 - There is a remainder. We can't divide by 101 evenly anymore. Let's try the next prime number
11,251 ÷ 103 = 109.233 - This has a remainder. 103 is not a factor.
11,251 ÷ 107 = 105.1495 - This has a remainder. 107 is not a factor.
11,251 ÷ 109 = 103.2202 - This has a remainder. 109 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
11,251 ÷ 11,251 = 1 - No remainder! 11,251 is one of the factors!

The orange divisor(s) above are the prime factors of the number 110,226,047. If we put all of it together we have the factors 97 x 101 x 11,251 = 110,226,047. It can also be written in exponential form as 97¹ x 101¹ x 11,251¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 110,226,047.

	110,226,047
97		1,136,351
	101		11,251

More Prime Factorization Examples

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