Q: What is the prime factorization of the number 137,401,413?

Why is the prime factorization of 137,401,413 written as 3¹ x 281¹ x 389¹ x 419¹?

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 137,401,413

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 137,401,413 by 2

137,401,413 ÷ 2 = 68,700,706.5 - This has a remainder. Let's try another prime number.
137,401,413 ÷ 3 = 45,800,471 - No remainder! 3 is one of the factors!
45,800,471 ÷ 3 = 15,266,823.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
45,800,471 ÷ 5 = 9,160,094.2 - This has a remainder. 5 is not a factor.
45,800,471 ÷ 7 = 6,542,924.4286 - This has a remainder. 7 is not a factor.
45,800,471 ÷ 11 = 4,163,679.1818 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
45,800,471 ÷ 281 = 162,991 - No remainder! 281 is one of the factors!
162,991 ÷ 281 = 580.0391 - There is a remainder. We can't divide by 281 evenly anymore. Let's try the next prime number
162,991 ÷ 283 = 575.9399 - This has a remainder. 283 is not a factor.
162,991 ÷ 293 = 556.2833 - This has a remainder. 293 is not a factor.
162,991 ÷ 307 = 530.9153 - This has a remainder. 307 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
162,991 ÷ 389 = 419 - No remainder! 389 is one of the factors!
419 ÷ 389 = 1.0771 - There is a remainder. We can't divide by 389 evenly anymore. Let's try the next prime number
419 ÷ 397 = 1.0554 - This has a remainder. 397 is not a factor.
419 ÷ 401 = 1.0449 - This has a remainder. 401 is not a factor.
419 ÷ 409 = 1.0244 - This has a remainder. 409 is not a factor.
419 ÷ 419 = 1 - No remainder! 419 is one of the factors!

The orange divisor(s) above are the prime factors of the number 137,401,413. If we put all of it together we have the factors 3 x 281 x 389 x 419 = 137,401,413. It can also be written in exponential form as 3¹ x 281¹ x 389¹ x 419¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 137,401,413.

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