Q: What is the prime factorization of the number 49,127,104?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 13 x 137 x 431
    • or also written as { 2, 2, 2, 2, 2, 2, 13, 137, 431 }
  • Written in exponential form: 26 x 131 x 1371 x 4311

Why is the prime factorization of 49,127,104 written as 26 x 131 x 1371 x 4311?

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 49,127,104

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 49,127,104 by 2

49,127,104 ÷ 2 = 24,563,552 - No remainder! 2 is one of the factors!
24,563,552 ÷ 2 = 12,281,776 - No remainder! 2 is one of the factors!
12,281,776 ÷ 2 = 6,140,888 - No remainder! 2 is one of the factors!
6,140,888 ÷ 2 = 3,070,444 - No remainder! 2 is one of the factors!
3,070,444 ÷ 2 = 1,535,222 - No remainder! 2 is one of the factors!
1,535,222 ÷ 2 = 767,611 - No remainder! 2 is one of the factors!
767,611 ÷ 2 = 383,805.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
767,611 ÷ 3 = 255,870.3333 - This has a remainder. 3 is not a factor.
767,611 ÷ 5 = 153,522.2 - This has a remainder. 5 is not a factor.
767,611 ÷ 7 = 109,658.7143 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
767,611 ÷ 13 = 59,047 - No remainder! 13 is one of the factors!
59,047 ÷ 13 = 4,542.0769 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
59,047 ÷ 17 = 3,473.3529 - This has a remainder. 17 is not a factor.
59,047 ÷ 19 = 3,107.7368 - This has a remainder. 19 is not a factor.
59,047 ÷ 23 = 2,567.2609 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
59,047 ÷ 137 = 431 - No remainder! 137 is one of the factors!
431 ÷ 137 = 3.146 - There is a remainder. We can't divide by 137 evenly anymore. Let's try the next prime number
431 ÷ 139 = 3.1007 - This has a remainder. 139 is not a factor.
431 ÷ 149 = 2.8926 - This has a remainder. 149 is not a factor.
431 ÷ 151 = 2.8543 - This has a remainder. 151 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
431 ÷ 431 = 1 - No remainder! 431 is one of the factors!

The orange divisor(s) above are the prime factors of the number 49,127,104. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 13 x 137 x 431 = 49,127,104. It can also be written in exponential form as 26 x 131 x 1371 x 4311.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 49,127,104.

49,127,104
Factor Arrows
224,563,552
Factor Arrows
212,281,776
Factor Arrows
26,140,888
Factor Arrows
23,070,444
Factor Arrows
21,535,222
Factor Arrows
2767,611
Factor Arrows
1359,047
Factor Arrows
137431

More Prime Factorization Examples

49,127,10249,127,10349,127,10549,127,106
21 x 291 x 411 x 731 x 283132 x 191 x 231 x 12,491151 x 971 x 101,293121 x 31 x 72 x 167,0991

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