Q: What is the prime factorization of the number 624,145,059?

Why is the prime factorization of 624,145,059 written as 3² x 691¹ x 100,361¹?

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 624,145,059

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 624,145,059 by 2

624,145,059 ÷ 2 = 312,072,529.5 - This has a remainder. Let's try another prime number.
624,145,059 ÷ 3 = 208,048,353 - No remainder! 3 is one of the factors!
208,048,353 ÷ 3 = 69,349,451 - No remainder! 3 is one of the factors!
69,349,451 ÷ 3 = 23,116,483.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
69,349,451 ÷ 5 = 13,869,890.2 - This has a remainder. 5 is not a factor.
69,349,451 ÷ 7 = 9,907,064.4286 - This has a remainder. 7 is not a factor.
69,349,451 ÷ 11 = 6,304,495.5455 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
69,349,451 ÷ 691 = 100,361 - No remainder! 691 is one of the factors!
100,361 ÷ 691 = 145.2402 - There is a remainder. We can't divide by 691 evenly anymore. Let's try the next prime number
100,361 ÷ 701 = 143.1683 - This has a remainder. 701 is not a factor.
100,361 ÷ 709 = 141.5529 - This has a remainder. 709 is not a factor.
100,361 ÷ 719 = 139.5841 - This has a remainder. 719 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
100,361 ÷ 100,361 = 1 - No remainder! 100,361 is one of the factors!

The orange divisor(s) above are the prime factors of the number 624,145,059. If we put all of it together we have the factors 3 x 3 x 691 x 100,361 = 624,145,059. It can also be written in exponential form as 3² x 691¹ x 100,361¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 624,145,059.

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