Q: What is the prime factorization of the number 65,778,272?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 7 x 103 x 2,851
    • or also written as { 2, 2, 2, 2, 2, 7, 103, 2,851 }
  • Written in exponential form: 25 x 71 x 1031 x 2,8511

Why is the prime factorization of 65,778,272 written as 25 x 71 x 1031 x 2,8511?

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 65,778,272

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 65,778,272 by 2

65,778,272 ÷ 2 = 32,889,136 - No remainder! 2 is one of the factors!
32,889,136 ÷ 2 = 16,444,568 - No remainder! 2 is one of the factors!
16,444,568 ÷ 2 = 8,222,284 - No remainder! 2 is one of the factors!
8,222,284 ÷ 2 = 4,111,142 - No remainder! 2 is one of the factors!
4,111,142 ÷ 2 = 2,055,571 - No remainder! 2 is one of the factors!
2,055,571 ÷ 2 = 1,027,785.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
2,055,571 ÷ 3 = 685,190.3333 - This has a remainder. 3 is not a factor.
2,055,571 ÷ 5 = 411,114.2 - This has a remainder. 5 is not a factor.
2,055,571 ÷ 7 = 293,653 - No remainder! 7 is one of the factors!
293,653 ÷ 7 = 41,950.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
293,653 ÷ 11 = 26,695.7273 - This has a remainder. 11 is not a factor.
293,653 ÷ 13 = 22,588.6923 - This has a remainder. 13 is not a factor.
293,653 ÷ 17 = 17,273.7059 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
293,653 ÷ 103 = 2,851 - No remainder! 103 is one of the factors!
2,851 ÷ 103 = 27.6796 - There is a remainder. We can't divide by 103 evenly anymore. Let's try the next prime number
2,851 ÷ 107 = 26.6449 - This has a remainder. 107 is not a factor.
2,851 ÷ 109 = 26.156 - This has a remainder. 109 is not a factor.
2,851 ÷ 113 = 25.2301 - This has a remainder. 113 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,851 ÷ 2,851 = 1 - No remainder! 2,851 is one of the factors!

The orange divisor(s) above are the prime factors of the number 65,778,272. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 7 x 103 x 2,851 = 65,778,272. It can also be written in exponential form as 25 x 71 x 1031 x 2,8511.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 65,778,272.

65,778,272
Factor Arrows
232,889,136
Factor Arrows
216,444,568
Factor Arrows
28,222,284
Factor Arrows
24,111,142
Factor Arrows
22,055,571
Factor Arrows
7293,653
Factor Arrows
1032,851

More Prime Factorization Examples

65,778,27065,778,27165,778,27365,778,274
21 x 31 x 51 x 171 x 1011 x 1,2771131 x 5,059,867132 x 111 x 664,427121 x 591 x 557,4431

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