Q: What is the prime factorization of the number 20,012,025?

 A:
  • The prime factors are: 3 x 5 x 5 x 11 x 127 x 191
    • or also written as { 3, 5, 5, 11, 127, 191 }
  • Written in exponential form: 31 x 52 x 111 x 1271 x 1911

Why is the prime factorization of 20,012,025 written as 31 x 52 x 111 x 1271 x 1911?

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 20,012,025

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.

If it doesn't make sense yet, let's try it...

Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Let's start by dividing 20,012,025 by 2

20,012,025 ÷ 2 = 10,006,012.5 - This has a remainder. Let's try another prime number.
20,012,025 ÷ 3 = 6,670,675 - No remainder! 3 is one of the factors!
6,670,675 ÷ 3 = 2,223,558.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
6,670,675 ÷ 5 = 1,334,135 - No remainder! 5 is one of the factors!
1,334,135 ÷ 5 = 266,827 - No remainder! 5 is one of the factors!
266,827 ÷ 5 = 53,365.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
266,827 ÷ 7 = 38,118.1429 - This has a remainder. 7 is not a factor.
266,827 ÷ 11 = 24,257 - No remainder! 11 is one of the factors!
24,257 ÷ 11 = 2,205.1818 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
24,257 ÷ 13 = 1,865.9231 - This has a remainder. 13 is not a factor.
24,257 ÷ 17 = 1,426.8824 - This has a remainder. 17 is not a factor.
24,257 ÷ 19 = 1,276.6842 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
24,257 ÷ 127 = 191 - No remainder! 127 is one of the factors!
191 ÷ 127 = 1.5039 - There is a remainder. We can't divide by 127 evenly anymore. Let's try the next prime number
191 ÷ 131 = 1.458 - This has a remainder. 131 is not a factor.
191 ÷ 137 = 1.3942 - This has a remainder. 137 is not a factor.
191 ÷ 139 = 1.3741 - This has a remainder. 139 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
191 ÷ 191 = 1 - No remainder! 191 is one of the factors!

The orange divisor(s) above are the prime factors of the number 20,012,025. If we put all of it together we have the factors 3 x 5 x 5 x 11 x 127 x 191 = 20,012,025. It can also be written in exponential form as 31 x 52 x 111 x 1271 x 1911.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 20,012,025.

20,012,025
Factor Arrows
36,670,675
Factor Arrows
51,334,135
Factor Arrows
5266,827
Factor Arrows
1124,257
Factor Arrows
127191

More Prime Factorization Examples

20,012,02320,012,02420,012,02620,012,027
20,012,023123 x 231 x 108,761121 x 171 x 611 x 9,649171 x 1,4091 x 2,0291

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