Q: What is the prime factorization of the number 312,023,101?

Why is the prime factorization of 312,023,101 written as 13¹ x 97¹ x 349¹ x 709¹?

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 312,023,101

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 312,023,101 by 2

312,023,101 ÷ 2 = 156,011,550.5 - This has a remainder. Let's try another prime number.
312,023,101 ÷ 3 = 104,007,700.3333 - This has a remainder. Let's try another prime number.
312,023,101 ÷ 5 = 62,404,620.2 - This has a remainder. Let's try another prime number.
312,023,101 ÷ 13 = 24,001,777 - No remainder! 13 is one of the factors!
24,001,777 ÷ 13 = 1,846,290.5385 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
24,001,777 ÷ 17 = 1,411,869.2353 - This has a remainder. 17 is not a factor.
24,001,777 ÷ 19 = 1,263,251.4211 - This has a remainder. 19 is not a factor.
24,001,777 ÷ 23 = 1,043,555.5217 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
24,001,777 ÷ 97 = 247,441 - No remainder! 97 is one of the factors!
247,441 ÷ 97 = 2,550.9381 - There is a remainder. We can't divide by 97 evenly anymore. Let's try the next prime number
247,441 ÷ 101 = 2,449.9109 - This has a remainder. 101 is not a factor.
247,441 ÷ 103 = 2,402.3398 - This has a remainder. 103 is not a factor.
247,441 ÷ 107 = 2,312.5327 - This has a remainder. 107 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
247,441 ÷ 349 = 709 - No remainder! 349 is one of the factors!
709 ÷ 349 = 2.0315 - There is a remainder. We can't divide by 349 evenly anymore. Let's try the next prime number
709 ÷ 353 = 2.0085 - This has a remainder. 353 is not a factor.
709 ÷ 359 = 1.9749 - This has a remainder. 359 is not a factor.
709 ÷ 367 = 1.9319 - This has a remainder. 367 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
709 ÷ 709 = 1 - No remainder! 709 is one of the factors!

The orange divisor(s) above are the prime factors of the number 312,023,101. If we put all of it together we have the factors 13 x 97 x 349 x 709 = 312,023,101. It can also be written in exponential form as 13¹ x 97¹ x 349¹ x 709¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 312,023,101.

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