Q: What is the prime factorization of the number 578,462,261?

Why is the prime factorization of 578,462,261 written as 13¹ x 139¹ x 157¹ x 2,039¹?

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 578,462,261

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 578,462,261 by 2

578,462,261 ÷ 2 = 289,231,130.5 - This has a remainder. Let's try another prime number.
578,462,261 ÷ 3 = 192,820,753.6667 - This has a remainder. Let's try another prime number.
578,462,261 ÷ 5 = 115,692,452.2 - This has a remainder. Let's try another prime number.
578,462,261 ÷ 13 = 44,497,097 - No remainder! 13 is one of the factors!
44,497,097 ÷ 13 = 3,422,853.6154 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
44,497,097 ÷ 17 = 2,617,476.2941 - This has a remainder. 17 is not a factor.
44,497,097 ÷ 19 = 2,341,952.4737 - This has a remainder. 19 is not a factor.
44,497,097 ÷ 23 = 1,934,656.3913 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
44,497,097 ÷ 139 = 320,123 - No remainder! 139 is one of the factors!
320,123 ÷ 139 = 2,303.0432 - There is a remainder. We can't divide by 139 evenly anymore. Let's try the next prime number
320,123 ÷ 149 = 2,148.4765 - This has a remainder. 149 is not a factor.
320,123 ÷ 151 = 2,120.0199 - This has a remainder. 151 is not a factor.
320,123 ÷ 157 = 2,039 - No remainder! 157 is one of the factors!
2,039 ÷ 157 = 12.9873 - There is a remainder. We can't divide by 157 evenly anymore. Let's try the next prime number
2,039 ÷ 163 = 12.5092 - This has a remainder. 163 is not a factor.
2,039 ÷ 167 = 12.2096 - This has a remainder. 167 is not a factor.
2,039 ÷ 173 = 11.7861 - This has a remainder. 173 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,039 ÷ 2,039 = 1 - No remainder! 2,039 is one of the factors!

The orange divisor(s) above are the prime factors of the number 578,462,261. If we put all of it together we have the factors 13 x 139 x 157 x 2,039 = 578,462,261. It can also be written in exponential form as 13¹ x 139¹ x 157¹ x 2,039¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 578,462,261.

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