Q: What is the prime factorization of the number 8,961,721?

 A:
  • The prime factors are: 103 x 167 x 521
    • or also written as { 103, 167, 521 }
  • Written in exponential form: 1031 x 1671 x 5211

Why is the prime factorization of 8,961,721 written as 1031 x 1671 x 5211?

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 8,961,721

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 8,961,721 by 2

8,961,721 ÷ 2 = 4,480,860.5 - This has a remainder. Let's try another prime number.
8,961,721 ÷ 3 = 2,987,240.3333 - This has a remainder. Let's try another prime number.
8,961,721 ÷ 5 = 1,792,344.2 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
8,961,721 ÷ 103 = 87,007 - No remainder! 103 is one of the factors!
87,007 ÷ 103 = 844.7282 - There is a remainder. We can't divide by 103 evenly anymore. Let's try the next prime number
87,007 ÷ 107 = 813.1495 - This has a remainder. 107 is not a factor.
87,007 ÷ 109 = 798.2294 - This has a remainder. 109 is not a factor.
87,007 ÷ 113 = 769.9735 - This has a remainder. 113 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
87,007 ÷ 167 = 521 - No remainder! 167 is one of the factors!
521 ÷ 167 = 3.1198 - There is a remainder. We can't divide by 167 evenly anymore. Let's try the next prime number
521 ÷ 173 = 3.0116 - This has a remainder. 173 is not a factor.
521 ÷ 179 = 2.9106 - This has a remainder. 179 is not a factor.
521 ÷ 181 = 2.8785 - This has a remainder. 181 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
521 ÷ 521 = 1 - No remainder! 521 is one of the factors!

The orange divisor(s) above are the prime factors of the number 8,961,721. If we put all of it together we have the factors 103 x 167 x 521 = 8,961,721. It can also be written in exponential form as 1031 x 1671 x 5211.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 8,961,721.

8,961,721
Factor Arrows
10387,007
Factor Arrows
167521

More Prime Factorization Examples

8,961,7198,961,7208,961,7228,961,723
131 x 671 x 10,289123 x 31 x 51 x 171 x 231 x 191121 x 71 x 111 x 58,193132 x 995,7471

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