Q: What is the prime factorization of the number 3,246,978?

 A:
  • The prime factors are: 2 x 3 x 7 x 97 x 797
    • or also written as { 2, 3, 7, 97, 797 }
  • Written in exponential form: 21 x 31 x 71 x 971 x 7971

Why is the prime factorization of 3,246,978 written as 21 x 31 x 71 x 971 x 7971?

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 3,246,978

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 3,246,978 by 2

3,246,978 ÷ 2 = 1,623,489 - No remainder! 2 is one of the factors!
1,623,489 ÷ 2 = 811,744.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,623,489 ÷ 3 = 541,163 - No remainder! 3 is one of the factors!
541,163 ÷ 3 = 180,387.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
541,163 ÷ 5 = 108,232.6 - This has a remainder. 5 is not a factor.
541,163 ÷ 7 = 77,309 - No remainder! 7 is one of the factors!
77,309 ÷ 7 = 11,044.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
77,309 ÷ 11 = 7,028.0909 - This has a remainder. 11 is not a factor.
77,309 ÷ 13 = 5,946.8462 - This has a remainder. 13 is not a factor.
77,309 ÷ 17 = 4,547.5882 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
77,309 ÷ 97 = 797 - No remainder! 97 is one of the factors!
797 ÷ 97 = 8.2165 - There is a remainder. We can't divide by 97 evenly anymore. Let's try the next prime number
797 ÷ 101 = 7.8911 - This has a remainder. 101 is not a factor.
797 ÷ 103 = 7.7379 - This has a remainder. 103 is not a factor.
797 ÷ 107 = 7.4486 - This has a remainder. 107 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
797 ÷ 797 = 1 - No remainder! 797 is one of the factors!

The orange divisor(s) above are the prime factors of the number 3,246,978. If we put all of it together we have the factors 2 x 3 x 7 x 97 x 797 = 3,246,978. It can also be written in exponential form as 21 x 31 x 71 x 971 x 7971.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 3,246,978.

3,246,978
Factor Arrows
21,623,489
Factor Arrows
3541,163
Factor Arrows
777,309
Factor Arrows
97797

More Prime Factorization Examples

3,246,9763,246,9773,246,9793,246,980
27 x 25,36713,246,9771231 x 791 x 1,787122 x 51 x 111 x 14,7591

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