Q: What is the prime factorization of the number 37,754,560?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 5 x 127 x 929
    • or also written as { 2, 2, 2, 2, 2, 2, 5, 127, 929 }
  • Written in exponential form: 26 x 51 x 1271 x 9291

Why is the prime factorization of 37,754,560 written as 26 x 51 x 1271 x 9291?

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 37,754,560

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 37,754,560 by 2

37,754,560 ÷ 2 = 18,877,280 - No remainder! 2 is one of the factors!
18,877,280 ÷ 2 = 9,438,640 - No remainder! 2 is one of the factors!
9,438,640 ÷ 2 = 4,719,320 - No remainder! 2 is one of the factors!
4,719,320 ÷ 2 = 2,359,660 - No remainder! 2 is one of the factors!
2,359,660 ÷ 2 = 1,179,830 - No remainder! 2 is one of the factors!
1,179,830 ÷ 2 = 589,915 - No remainder! 2 is one of the factors!
589,915 ÷ 2 = 294,957.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
589,915 ÷ 3 = 196,638.3333 - This has a remainder. 3 is not a factor.
589,915 ÷ 5 = 117,983 - No remainder! 5 is one of the factors!
117,983 ÷ 5 = 23,596.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
117,983 ÷ 7 = 16,854.7143 - This has a remainder. 7 is not a factor.
117,983 ÷ 11 = 10,725.7273 - This has a remainder. 11 is not a factor.
117,983 ÷ 13 = 9,075.6154 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
117,983 ÷ 127 = 929 - No remainder! 127 is one of the factors!
929 ÷ 127 = 7.315 - There is a remainder. We can't divide by 127 evenly anymore. Let's try the next prime number
929 ÷ 131 = 7.0916 - This has a remainder. 131 is not a factor.
929 ÷ 137 = 6.781 - This has a remainder. 137 is not a factor.
929 ÷ 139 = 6.6835 - This has a remainder. 139 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
929 ÷ 929 = 1 - No remainder! 929 is one of the factors!

The orange divisor(s) above are the prime factors of the number 37,754,560. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 5 x 127 x 929 = 37,754,560. It can also be written in exponential form as 26 x 51 x 1271 x 9291.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 37,754,560.

37,754,560
Factor Arrows
218,877,280
Factor Arrows
29,438,640
Factor Arrows
24,719,320
Factor Arrows
22,359,660
Factor Arrows
21,179,830
Factor Arrows
2589,915
Factor Arrows
5117,983
Factor Arrows
127929

More Prime Factorization Examples

37,754,55837,754,55937,754,56137,754,562
21 x 191 x 993,541133 x 311 x 431 x 1,0491131 x 2,904,197121 x 31 x 6,292,4271

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