Q: What is the prime factorization of the number 420,141,527?

Why is the prime factorization of 420,141,527 written as 13¹ x 127¹ x 397¹ x 641¹?

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 420,141,527

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 420,141,527 by 2

420,141,527 ÷ 2 = 210,070,763.5 - This has a remainder. Let's try another prime number.
420,141,527 ÷ 3 = 140,047,175.6667 - This has a remainder. Let's try another prime number.
420,141,527 ÷ 5 = 84,028,305.4 - This has a remainder. Let's try another prime number.
420,141,527 ÷ 13 = 32,318,579 - No remainder! 13 is one of the factors!
32,318,579 ÷ 13 = 2,486,044.5385 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
32,318,579 ÷ 17 = 1,901,092.8824 - This has a remainder. 17 is not a factor.
32,318,579 ÷ 19 = 1,700,977.8421 - This has a remainder. 19 is not a factor.
32,318,579 ÷ 23 = 1,405,155.6087 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
32,318,579 ÷ 127 = 254,477 - No remainder! 127 is one of the factors!
254,477 ÷ 127 = 2,003.7559 - There is a remainder. We can't divide by 127 evenly anymore. Let's try the next prime number
254,477 ÷ 131 = 1,942.5725 - This has a remainder. 131 is not a factor.
254,477 ÷ 137 = 1,857.4964 - This has a remainder. 137 is not a factor.
254,477 ÷ 139 = 1,830.7698 - This has a remainder. 139 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
254,477 ÷ 397 = 641 - No remainder! 397 is one of the factors!
641 ÷ 397 = 1.6146 - There is a remainder. We can't divide by 397 evenly anymore. Let's try the next prime number
641 ÷ 401 = 1.5985 - This has a remainder. 401 is not a factor.
641 ÷ 409 = 1.5672 - This has a remainder. 409 is not a factor.
641 ÷ 419 = 1.5298 - This has a remainder. 419 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
641 ÷ 641 = 1 - No remainder! 641 is one of the factors!

The orange divisor(s) above are the prime factors of the number 420,141,527. If we put all of it together we have the factors 13 x 127 x 397 x 641 = 420,141,527. It can also be written in exponential form as 13¹ x 127¹ x 397¹ x 641¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 420,141,527.

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