Q: What is the prime factorization of the number 85,176,126?

 A:
  • The prime factors are: 2 x 3 x 3 x 7 x 19 x 47 x 757
    • or also written as { 2, 3, 3, 7, 19, 47, 757 }
  • Written in exponential form: 21 x 32 x 71 x 191 x 471 x 7571

Why is the prime factorization of 85,176,126 written as 21 x 32 x 71 x 191 x 471 x 7571?

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 85,176,126

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 85,176,126 by 2

85,176,126 ÷ 2 = 42,588,063 - No remainder! 2 is one of the factors!
42,588,063 ÷ 2 = 21,294,031.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
42,588,063 ÷ 3 = 14,196,021 - No remainder! 3 is one of the factors!
14,196,021 ÷ 3 = 4,732,007 - No remainder! 3 is one of the factors!
4,732,007 ÷ 3 = 1,577,335.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
4,732,007 ÷ 5 = 946,401.4 - This has a remainder. 5 is not a factor.
4,732,007 ÷ 7 = 676,001 - No remainder! 7 is one of the factors!
676,001 ÷ 7 = 96,571.5714 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
676,001 ÷ 11 = 61,454.6364 - This has a remainder. 11 is not a factor.
676,001 ÷ 13 = 52,000.0769 - This has a remainder. 13 is not a factor.
676,001 ÷ 17 = 39,764.7647 - This has a remainder. 17 is not a factor.
676,001 ÷ 19 = 35,579 - No remainder! 19 is one of the factors!
35,579 ÷ 19 = 1,872.5789 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
35,579 ÷ 23 = 1,546.913 - This has a remainder. 23 is not a factor.
35,579 ÷ 29 = 1,226.8621 - This has a remainder. 29 is not a factor.
35,579 ÷ 31 = 1,147.7097 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
35,579 ÷ 47 = 757 - No remainder! 47 is one of the factors!
757 ÷ 47 = 16.1064 - There is a remainder. We can't divide by 47 evenly anymore. Let's try the next prime number
757 ÷ 53 = 14.283 - This has a remainder. 53 is not a factor.
757 ÷ 59 = 12.8305 - This has a remainder. 59 is not a factor.
757 ÷ 61 = 12.4098 - This has a remainder. 61 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
757 ÷ 757 = 1 - No remainder! 757 is one of the factors!

The orange divisor(s) above are the prime factors of the number 85,176,126. If we put all of it together we have the factors 2 x 3 x 3 x 7 x 19 x 47 x 757 = 85,176,126. It can also be written in exponential form as 21 x 32 x 71 x 191 x 471 x 7571.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 85,176,126.

85,176,126
Factor Arrows
242,588,063
Factor Arrows
314,196,021
Factor Arrows
34,732,007
Factor Arrows
7676,001
Factor Arrows
1935,579
Factor Arrows
47757

More Prime Factorization Examples

85,176,12485,176,12585,176,12785,176,128
22 x 111 x 6011 x 3,221153 x 681,4091311 x 2,747,617126 x 1011 x 13,1771

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