Q: What is the prime factorization of the number 701,408,736?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7 x 23 x 2,161
    • or also written as { 2, 2, 2, 2, 2, 3, 3, 7, 7, 23, 2,161 }
  • Written in exponential form: 25 x 32 x 72 x 231 x 2,1611

Why is the prime factorization of 701,408,736 written as 25 x 32 x 72 x 231 x 2,1611?

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 701,408,736

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 701,408,736 by 2

701,408,736 ÷ 2 = 350,704,368 - No remainder! 2 is one of the factors!
350,704,368 ÷ 2 = 175,352,184 - No remainder! 2 is one of the factors!
175,352,184 ÷ 2 = 87,676,092 - No remainder! 2 is one of the factors!
87,676,092 ÷ 2 = 43,838,046 - No remainder! 2 is one of the factors!
43,838,046 ÷ 2 = 21,919,023 - No remainder! 2 is one of the factors!
21,919,023 ÷ 2 = 10,959,511.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
21,919,023 ÷ 3 = 7,306,341 - No remainder! 3 is one of the factors!
7,306,341 ÷ 3 = 2,435,447 - No remainder! 3 is one of the factors!
2,435,447 ÷ 3 = 811,815.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
2,435,447 ÷ 5 = 487,089.4 - This has a remainder. 5 is not a factor.
2,435,447 ÷ 7 = 347,921 - No remainder! 7 is one of the factors!
347,921 ÷ 7 = 49,703 - No remainder! 7 is one of the factors!
49,703 ÷ 7 = 7,100.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
49,703 ÷ 11 = 4,518.4545 - This has a remainder. 11 is not a factor.
49,703 ÷ 13 = 3,823.3077 - This has a remainder. 13 is not a factor.
49,703 ÷ 17 = 2,923.7059 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
49,703 ÷ 23 = 2,161 - No remainder! 23 is one of the factors!
2,161 ÷ 23 = 93.9565 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
2,161 ÷ 29 = 74.5172 - This has a remainder. 29 is not a factor.
2,161 ÷ 31 = 69.7097 - This has a remainder. 31 is not a factor.
2,161 ÷ 37 = 58.4054 - This has a remainder. 37 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,161 ÷ 2,161 = 1 - No remainder! 2,161 is one of the factors!

The orange divisor(s) above are the prime factors of the number 701,408,736. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7 x 23 x 2,161 = 701,408,736. It can also be written in exponential form as 25 x 32 x 72 x 231 x 2,1611.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 701,408,736.

701,408,736
Factor Arrows
2350,704,368
Factor Arrows
2175,352,184
Factor Arrows
287,676,092
Factor Arrows
243,838,046
Factor Arrows
221,919,023
Factor Arrows
37,306,341
Factor Arrows
32,435,447
Factor Arrows
7347,921
Factor Arrows
749,703
Factor Arrows
232,161

More Prime Factorization Examples

701,408,734701,408,735701,408,737701,408,738
21 x 131 x 1391 x 4211 x 461151 x 1,2291 x 114,14317091 x 989,293121 x 4391 x 798,8711

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