Q: What is the prime factorization of the number 415,643,410?

Why is the prime factorization of 415,643,410 written as 2¹ x 5¹ x 7¹ x 13¹ x 661¹ x 691¹?

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 415,643,410

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 415,643,410 by 2

415,643,410 ÷ 2 = 207,821,705 - No remainder! 2 is one of the factors!
207,821,705 ÷ 2 = 103,910,852.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
207,821,705 ÷ 3 = 69,273,901.6667 - This has a remainder. 3 is not a factor.
207,821,705 ÷ 5 = 41,564,341 - No remainder! 5 is one of the factors!
41,564,341 ÷ 5 = 8,312,868.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
41,564,341 ÷ 7 = 5,937,763 - No remainder! 7 is one of the factors!
5,937,763 ÷ 7 = 848,251.8571 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
5,937,763 ÷ 11 = 539,796.6364 - This has a remainder. 11 is not a factor.
5,937,763 ÷ 13 = 456,751 - No remainder! 13 is one of the factors!
456,751 ÷ 13 = 35,134.6923 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
456,751 ÷ 17 = 26,867.7059 - This has a remainder. 17 is not a factor.
456,751 ÷ 19 = 24,039.5263 - This has a remainder. 19 is not a factor.
456,751 ÷ 23 = 19,858.7391 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
456,751 ÷ 661 = 691 - No remainder! 661 is one of the factors!
691 ÷ 661 = 1.0454 - There is a remainder. We can't divide by 661 evenly anymore. Let's try the next prime number
691 ÷ 673 = 1.0267 - This has a remainder. 673 is not a factor.
691 ÷ 677 = 1.0207 - This has a remainder. 677 is not a factor.
691 ÷ 683 = 1.0117 - This has a remainder. 683 is not a factor.
691 ÷ 691 = 1 - No remainder! 691 is one of the factors!

The orange divisor(s) above are the prime factors of the number 415,643,410. If we put all of it together we have the factors 2 x 5 x 7 x 13 x 661 x 691 = 415,643,410. It can also be written in exponential form as 2¹ x 5¹ x 7¹ x 13¹ x 661¹ x 691¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 415,643,410.

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