Q: What is the prime factorization of the number 313,221,120?

 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 2 x 2 x 2 x 3 x 5 x 2,549
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 2,549 }
  • Written in exponential form: 213 x 31 x 51 x 2,5491

Why is the prime factorization of 313,221,120 written as 213 x 31 x 51 x 2,5491?

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 313,221,120

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 313,221,120 by 2

313,221,120 ÷ 2 = 156,610,560 - No remainder! 2 is one of the factors!
156,610,560 ÷ 2 = 78,305,280 - No remainder! 2 is one of the factors!
78,305,280 ÷ 2 = 39,152,640 - No remainder! 2 is one of the factors!
39,152,640 ÷ 2 = 19,576,320 - No remainder! 2 is one of the factors!
19,576,320 ÷ 2 = 9,788,160 - No remainder! 2 is one of the factors!
9,788,160 ÷ 2 = 4,894,080 - No remainder! 2 is one of the factors!
4,894,080 ÷ 2 = 2,447,040 - No remainder! 2 is one of the factors!
2,447,040 ÷ 2 = 1,223,520 - No remainder! 2 is one of the factors!
1,223,520 ÷ 2 = 611,760 - No remainder! 2 is one of the factors!
611,760 ÷ 2 = 305,880 - No remainder! 2 is one of the factors!
305,880 ÷ 2 = 152,940 - No remainder! 2 is one of the factors!
152,940 ÷ 2 = 76,470 - No remainder! 2 is one of the factors!
76,470 ÷ 2 = 38,235 - No remainder! 2 is one of the factors!
38,235 ÷ 2 = 19,117.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
38,235 ÷ 3 = 12,745 - No remainder! 3 is one of the factors!
12,745 ÷ 3 = 4,248.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
12,745 ÷ 5 = 2,549 - No remainder! 5 is one of the factors!
2,549 ÷ 5 = 509.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
2,549 ÷ 7 = 364.1429 - This has a remainder. 7 is not a factor.
2,549 ÷ 11 = 231.7273 - This has a remainder. 11 is not a factor.
2,549 ÷ 13 = 196.0769 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,549 ÷ 2,549 = 1 - No remainder! 2,549 is one of the factors!

The orange divisor(s) above are the prime factors of the number 313,221,120. 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 2 x 2 x 2 x 3 x 5 x 2,549 = 313,221,120. It can also be written in exponential form as 213 x 31 x 51 x 2,5491.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 313,221,120.

313,221,120
Factor Arrows
2156,610,560
Factor Arrows
278,305,280
Factor Arrows
239,152,640
Factor Arrows
219,576,320
Factor Arrows
29,788,160
Factor Arrows
24,894,080
Factor Arrows
22,447,040
Factor Arrows
21,223,520
Factor Arrows
2611,760
Factor Arrows
2305,880
Factor Arrows
2152,940
Factor Arrows
276,470
Factor Arrows
238,235
Factor Arrows
312,745
Factor Arrows
52,549

More Prime Factorization Examples

313,221,118313,221,119313,221,121313,221,122
21 x 71 x 191 x 5031 x 2,3411313,221,11912,7311 x 114,691121 x 1,0511 x 149,0111

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