Q: What is the prime factorization of the number 412,554,240?

 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 3 x 5 x 7 x 1,279
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 5, 7, 1,279 }
  • Written in exponential form: 210 x 32 x 51 x 71 x 1,2791

Why is the prime factorization of 412,554,240 written as 210 x 32 x 51 x 71 x 1,2791?

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 412,554,240

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 412,554,240 by 2

412,554,240 ÷ 2 = 206,277,120 - No remainder! 2 is one of the factors!
206,277,120 ÷ 2 = 103,138,560 - No remainder! 2 is one of the factors!
103,138,560 ÷ 2 = 51,569,280 - No remainder! 2 is one of the factors!
51,569,280 ÷ 2 = 25,784,640 - No remainder! 2 is one of the factors!
25,784,640 ÷ 2 = 12,892,320 - No remainder! 2 is one of the factors!
12,892,320 ÷ 2 = 6,446,160 - No remainder! 2 is one of the factors!
6,446,160 ÷ 2 = 3,223,080 - No remainder! 2 is one of the factors!
3,223,080 ÷ 2 = 1,611,540 - No remainder! 2 is one of the factors!
1,611,540 ÷ 2 = 805,770 - No remainder! 2 is one of the factors!
805,770 ÷ 2 = 402,885 - No remainder! 2 is one of the factors!
402,885 ÷ 2 = 201,442.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
402,885 ÷ 3 = 134,295 - No remainder! 3 is one of the factors!
134,295 ÷ 3 = 44,765 - No remainder! 3 is one of the factors!
44,765 ÷ 3 = 14,921.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
44,765 ÷ 5 = 8,953 - No remainder! 5 is one of the factors!
8,953 ÷ 5 = 1,790.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
8,953 ÷ 7 = 1,279 - No remainder! 7 is one of the factors!
1,279 ÷ 7 = 182.7143 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
1,279 ÷ 11 = 116.2727 - This has a remainder. 11 is not a factor.
1,279 ÷ 13 = 98.3846 - This has a remainder. 13 is not a factor.
1,279 ÷ 17 = 75.2353 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,279 ÷ 1,279 = 1 - No remainder! 1,279 is one of the factors!

The orange divisor(s) above are the prime factors of the number 412,554,240. 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 3 x 5 x 7 x 1,279 = 412,554,240. It can also be written in exponential form as 210 x 32 x 51 x 71 x 1,2791.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 412,554,240.

412,554,240
Factor Arrows
2206,277,120
Factor Arrows
2103,138,560
Factor Arrows
251,569,280
Factor Arrows
225,784,640
Factor Arrows
212,892,320
Factor Arrows
26,446,160
Factor Arrows
23,223,080
Factor Arrows
21,611,540
Factor Arrows
2805,770
Factor Arrows
2402,885
Factor Arrows
3134,295
Factor Arrows
344,765
Factor Arrows
58,953
Factor Arrows
71,279

More Prime Factorization Examples

412,554,238412,554,239412,554,241412,554,242
21 x 1571 x 4991 x 2,6331191 x 831 x 1311 x 1,9971111 x 37,504,931121 x 671 x 3671 x 8,3891

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