Q: What is the prime factorization of the number 421,526,560?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 5 x 7 x 13 x 13 x 17 x 131
    • or also written as { 2, 2, 2, 2, 2, 5, 7, 13, 13, 17, 131 }
  • Written in exponential form: 25 x 51 x 71 x 132 x 171 x 1311

Why is the prime factorization of 421,526,560 written as 25 x 51 x 71 x 132 x 171 x 1311?

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 421,526,560

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 421,526,560 by 2

421,526,560 ÷ 2 = 210,763,280 - No remainder! 2 is one of the factors!
210,763,280 ÷ 2 = 105,381,640 - No remainder! 2 is one of the factors!
105,381,640 ÷ 2 = 52,690,820 - No remainder! 2 is one of the factors!
52,690,820 ÷ 2 = 26,345,410 - No remainder! 2 is one of the factors!
26,345,410 ÷ 2 = 13,172,705 - No remainder! 2 is one of the factors!
13,172,705 ÷ 2 = 6,586,352.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
13,172,705 ÷ 3 = 4,390,901.6667 - This has a remainder. 3 is not a factor.
13,172,705 ÷ 5 = 2,634,541 - No remainder! 5 is one of the factors!
2,634,541 ÷ 5 = 526,908.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
2,634,541 ÷ 7 = 376,363 - No remainder! 7 is one of the factors!
376,363 ÷ 7 = 53,766.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
376,363 ÷ 11 = 34,214.8182 - This has a remainder. 11 is not a factor.
376,363 ÷ 13 = 28,951 - No remainder! 13 is one of the factors!
28,951 ÷ 13 = 2,227 - No remainder! 13 is one of the factors!
2,227 ÷ 13 = 171.3077 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
2,227 ÷ 17 = 131 - No remainder! 17 is one of the factors!
131 ÷ 17 = 7.7059 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
131 ÷ 19 = 6.8947 - This has a remainder. 19 is not a factor.
131 ÷ 23 = 5.6957 - This has a remainder. 23 is not a factor.
131 ÷ 29 = 4.5172 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
131 ÷ 131 = 1 - No remainder! 131 is one of the factors!

The orange divisor(s) above are the prime factors of the number 421,526,560. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 5 x 7 x 13 x 13 x 17 x 131 = 421,526,560. It can also be written in exponential form as 25 x 51 x 71 x 132 x 171 x 1311.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 421,526,560.

421,526,560
Factor Arrows
2210,763,280
Factor Arrows
2105,381,640
Factor Arrows
252,690,820
Factor Arrows
226,345,410
Factor Arrows
213,172,705
Factor Arrows
52,634,541
Factor Arrows
7376,363
Factor Arrows
1328,951
Factor Arrows
132,227
Factor Arrows
17131

More Prime Factorization Examples

421,526,558421,526,559421,526,561421,526,562
21 x 210,763,279131 x 1,7091 x 82,2171311 x 13,597,631121 x 31 x 591 x 1,190,7531

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