Q: What is the prime factorization of the number 123,115,464?

 A:
  • The prime factors are: 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 63,331
    • or also written as { 2, 2, 2, 3, 3, 3, 3, 3, 63,331 }
  • Written in exponential form: 23 x 35 x 63,3311

Why is the prime factorization of 123,115,464 written as 23 x 35 x 63,3311?

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 123,115,464

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 123,115,464 by 2

123,115,464 ÷ 2 = 61,557,732 - No remainder! 2 is one of the factors!
61,557,732 ÷ 2 = 30,778,866 - No remainder! 2 is one of the factors!
30,778,866 ÷ 2 = 15,389,433 - No remainder! 2 is one of the factors!
15,389,433 ÷ 2 = 7,694,716.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
15,389,433 ÷ 3 = 5,129,811 - No remainder! 3 is one of the factors!
5,129,811 ÷ 3 = 1,709,937 - No remainder! 3 is one of the factors!
1,709,937 ÷ 3 = 569,979 - No remainder! 3 is one of the factors!
569,979 ÷ 3 = 189,993 - No remainder! 3 is one of the factors!
189,993 ÷ 3 = 63,331 - No remainder! 3 is one of the factors!
63,331 ÷ 3 = 21,110.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
63,331 ÷ 5 = 12,666.2 - This has a remainder. 5 is not a factor.
63,331 ÷ 7 = 9,047.2857 - This has a remainder. 7 is not a factor.
63,331 ÷ 11 = 5,757.3636 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
63,331 ÷ 63,331 = 1 - No remainder! 63,331 is one of the factors!

The orange divisor(s) above are the prime factors of the number 123,115,464. If we put all of it together we have the factors 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 63,331 = 123,115,464. It can also be written in exponential form as 23 x 35 x 63,3311.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 123,115,464.

123,115,464
Factor Arrows
261,557,732
Factor Arrows
230,778,866
Factor Arrows
215,389,433
Factor Arrows
35,129,811
Factor Arrows
31,709,937
Factor Arrows
3569,979
Factor Arrows
3189,993
Factor Arrows
363,331

More Prime Factorization Examples

123,115,462123,115,463123,115,465123,115,466
21 x 171 x 3,621,04313,5831 x 34,361151 x 111 x 371 x 1011 x 599121 x 471 x 1,309,7391

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