Q: What is the prime factorization of the number 140,574,720?

 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 2 x 2 x 2 x 3 x 5 x 11 x 13
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 11, 13 }
  • Written in exponential form: 216 x 31 x 51 x 111 x 131

Why is the prime factorization of 140,574,720 written as 216 x 31 x 51 x 111 x 131?

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 140,574,720

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 140,574,720 by 2

140,574,720 ÷ 2 = 70,287,360 - No remainder! 2 is one of the factors!
70,287,360 ÷ 2 = 35,143,680 - No remainder! 2 is one of the factors!
35,143,680 ÷ 2 = 17,571,840 - No remainder! 2 is one of the factors!
17,571,840 ÷ 2 = 8,785,920 - No remainder! 2 is one of the factors!
8,785,920 ÷ 2 = 4,392,960 - No remainder! 2 is one of the factors!
4,392,960 ÷ 2 = 2,196,480 - No remainder! 2 is one of the factors!
2,196,480 ÷ 2 = 1,098,240 - No remainder! 2 is one of the factors!
1,098,240 ÷ 2 = 549,120 - No remainder! 2 is one of the factors!
549,120 ÷ 2 = 274,560 - No remainder! 2 is one of the factors!
274,560 ÷ 2 = 137,280 - No remainder! 2 is one of the factors!
137,280 ÷ 2 = 68,640 - No remainder! 2 is one of the factors!
68,640 ÷ 2 = 34,320 - No remainder! 2 is one of the factors!
34,320 ÷ 2 = 17,160 - No remainder! 2 is one of the factors!
17,160 ÷ 2 = 8,580 - No remainder! 2 is one of the factors!
8,580 ÷ 2 = 4,290 - No remainder! 2 is one of the factors!
4,290 ÷ 2 = 2,145 - No remainder! 2 is one of the factors!
2,145 ÷ 2 = 1,072.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
2,145 ÷ 3 = 715 - No remainder! 3 is one of the factors!
715 ÷ 3 = 238.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
715 ÷ 5 = 143 - No remainder! 5 is one of the factors!
143 ÷ 5 = 28.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
143 ÷ 7 = 20.4286 - This has a remainder. 7 is not a factor.
143 ÷ 11 = 13 - No remainder! 11 is one of the factors!
13 ÷ 11 = 1.1818 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
13 ÷ 13 = 1 - No remainder! 13 is one of the factors!

The orange divisor(s) above are the prime factors of the number 140,574,720. 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 2 x 2 x 2 x 3 x 5 x 11 x 13 = 140,574,720. It can also be written in exponential form as 216 x 31 x 51 x 111 x 131.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 140,574,720.

140,574,720
Factor Arrows
270,287,360
Factor Arrows
235,143,680
Factor Arrows
217,571,840
Factor Arrows
28,785,920
Factor Arrows
24,392,960
Factor Arrows
22,196,480
Factor Arrows
21,098,240
Factor Arrows
2549,120
Factor Arrows
2274,560
Factor Arrows
2137,280
Factor Arrows
268,640
Factor Arrows
234,320
Factor Arrows
217,160
Factor Arrows
28,580
Factor Arrows
24,290
Factor Arrows
22,145
Factor Arrows
3715
Factor Arrows
5143
Factor Arrows
1113

More Prime Factorization Examples

140,574,718140,574,719140,574,721140,574,722
21 x 70,287,3591140,574,719171 x 1,4991 x 13,397121 x 1,7211 x 40,8411

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