Q: What is the prime factorization of the number 111,682,560?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 11 x 661
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 11, 661 }
  • Written in exponential form: 210 x 31 x 51 x 111 x 6611

Why is the prime factorization of 111,682,560 written as 210 x 31 x 51 x 111 x 6611?

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 111,682,560

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 111,682,560 by 2

111,682,560 ÷ 2 = 55,841,280 - No remainder! 2 is one of the factors!
55,841,280 ÷ 2 = 27,920,640 - No remainder! 2 is one of the factors!
27,920,640 ÷ 2 = 13,960,320 - No remainder! 2 is one of the factors!
13,960,320 ÷ 2 = 6,980,160 - No remainder! 2 is one of the factors!
6,980,160 ÷ 2 = 3,490,080 - No remainder! 2 is one of the factors!
3,490,080 ÷ 2 = 1,745,040 - No remainder! 2 is one of the factors!
1,745,040 ÷ 2 = 872,520 - No remainder! 2 is one of the factors!
872,520 ÷ 2 = 436,260 - No remainder! 2 is one of the factors!
436,260 ÷ 2 = 218,130 - No remainder! 2 is one of the factors!
218,130 ÷ 2 = 109,065 - No remainder! 2 is one of the factors!
109,065 ÷ 2 = 54,532.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
109,065 ÷ 3 = 36,355 - No remainder! 3 is one of the factors!
36,355 ÷ 3 = 12,118.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
36,355 ÷ 5 = 7,271 - No remainder! 5 is one of the factors!
7,271 ÷ 5 = 1,454.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
7,271 ÷ 7 = 1,038.7143 - This has a remainder. 7 is not a factor.
7,271 ÷ 11 = 661 - No remainder! 11 is one of the factors!
661 ÷ 11 = 60.0909 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
661 ÷ 13 = 50.8462 - This has a remainder. 13 is not a factor.
661 ÷ 17 = 38.8824 - This has a remainder. 17 is not a factor.
661 ÷ 19 = 34.7895 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
661 ÷ 661 = 1 - No remainder! 661 is one of the factors!

The orange divisor(s) above are the prime factors of the number 111,682,560. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 11 x 661 = 111,682,560. It can also be written in exponential form as 210 x 31 x 51 x 111 x 6611.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 111,682,560.

111,682,560
Factor Arrows
255,841,280
Factor Arrows
227,920,640
Factor Arrows
213,960,320
Factor Arrows
26,980,160
Factor Arrows
23,490,080
Factor Arrows
21,745,040
Factor Arrows
2872,520
Factor Arrows
2436,260
Factor Arrows
2218,130
Factor Arrows
2109,065
Factor Arrows
336,355
Factor Arrows
57,271
Factor Arrows
11661

More Prime Factorization Examples

111,682,558111,682,559111,682,561111,682,562
21 x 131 x 1,6011 x 2,6831831 x 7431 x 1,8111111,682,561121 x 9831 x 56,8071

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