Q: What is the prime factorization of the number 65,251,801?

Why is the prime factorization of 65,251,801 written as 433¹ x 150,697¹?

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 65,251,801

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 65,251,801 by 2

65,251,801 ÷ 2 = 32,625,900.5 - This has a remainder. Let's try another prime number.
65,251,801 ÷ 3 = 21,750,600.3333 - This has a remainder. Let's try another prime number.
65,251,801 ÷ 5 = 13,050,360.2 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
65,251,801 ÷ 433 = 150,697 - No remainder! 433 is one of the factors!
150,697 ÷ 433 = 348.03 - There is a remainder. We can't divide by 433 evenly anymore. Let's try the next prime number
150,697 ÷ 439 = 343.2733 - This has a remainder. 439 is not a factor.
150,697 ÷ 443 = 340.1738 - This has a remainder. 443 is not a factor.
150,697 ÷ 449 = 335.6281 - This has a remainder. 449 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
150,697 ÷ 150,697 = 1 - No remainder! 150,697 is one of the factors!

The orange divisor(s) above are the prime factors of the number 65,251,801. If we put all of it together we have the factors 433 x 150,697 = 65,251,801. It can also be written in exponential form as 433¹ x 150,697¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 65,251,801.

	65,251,801
433		150,697

More Prime Factorization Examples

Try the factor calculator