Q: What is the prime factorization of the number 211,322,133?

Why is the prime factorization of 211,322,133 written as 3² x 11¹ x 31¹ x 37¹ x 1,861¹?

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 211,322,133

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 211,322,133 by 2

211,322,133 ÷ 2 = 105,661,066.5 - This has a remainder. Let's try another prime number.
211,322,133 ÷ 3 = 70,440,711 - No remainder! 3 is one of the factors!
70,440,711 ÷ 3 = 23,480,237 - No remainder! 3 is one of the factors!
23,480,237 ÷ 3 = 7,826,745.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
23,480,237 ÷ 5 = 4,696,047.4 - This has a remainder. 5 is not a factor.
23,480,237 ÷ 7 = 3,354,319.5714 - This has a remainder. 7 is not a factor.
23,480,237 ÷ 11 = 2,134,567 - No remainder! 11 is one of the factors!
2,134,567 ÷ 11 = 194,051.5455 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
2,134,567 ÷ 13 = 164,197.4615 - This has a remainder. 13 is not a factor.
2,134,567 ÷ 17 = 125,562.7647 - This has a remainder. 17 is not a factor.
2,134,567 ÷ 19 = 112,345.6316 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,134,567 ÷ 31 = 68,857 - No remainder! 31 is one of the factors!
68,857 ÷ 31 = 2,221.1935 - There is a remainder. We can't divide by 31 evenly anymore. Let's try the next prime number
68,857 ÷ 37 = 1,861 - No remainder! 37 is one of the factors!
1,861 ÷ 37 = 50.2973 - There is a remainder. We can't divide by 37 evenly anymore. Let's try the next prime number
1,861 ÷ 41 = 45.3902 - This has a remainder. 41 is not a factor.
1,861 ÷ 43 = 43.2791 - This has a remainder. 43 is not a factor.
1,861 ÷ 47 = 39.5957 - This has a remainder. 47 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,861 ÷ 1,861 = 1 - No remainder! 1,861 is one of the factors!

The orange divisor(s) above are the prime factors of the number 211,322,133. If we put all of it together we have the factors 3 x 3 x 11 x 31 x 37 x 1,861 = 211,322,133. It can also be written in exponential form as 3² x 11¹ x 31¹ x 37¹ x 1,861¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 211,322,133.

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