Q: What is the prime factorization of the number 42,501,120?

 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 3 x 5 x 2,767
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 2,767 }
  • Written in exponential form: 210 x 31 x 51 x 2,7671

Why is the prime factorization of 42,501,120 written as 210 x 31 x 51 x 2,7671?

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 42,501,120

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 42,501,120 by 2

42,501,120 ÷ 2 = 21,250,560 - No remainder! 2 is one of the factors!
21,250,560 ÷ 2 = 10,625,280 - No remainder! 2 is one of the factors!
10,625,280 ÷ 2 = 5,312,640 - No remainder! 2 is one of the factors!
5,312,640 ÷ 2 = 2,656,320 - No remainder! 2 is one of the factors!
2,656,320 ÷ 2 = 1,328,160 - No remainder! 2 is one of the factors!
1,328,160 ÷ 2 = 664,080 - No remainder! 2 is one of the factors!
664,080 ÷ 2 = 332,040 - No remainder! 2 is one of the factors!
332,040 ÷ 2 = 166,020 - No remainder! 2 is one of the factors!
166,020 ÷ 2 = 83,010 - No remainder! 2 is one of the factors!
83,010 ÷ 2 = 41,505 - No remainder! 2 is one of the factors!
41,505 ÷ 2 = 20,752.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
41,505 ÷ 3 = 13,835 - No remainder! 3 is one of the factors!
13,835 ÷ 3 = 4,611.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
13,835 ÷ 5 = 2,767 - No remainder! 5 is one of the factors!
2,767 ÷ 5 = 553.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
2,767 ÷ 7 = 395.2857 - This has a remainder. 7 is not a factor.
2,767 ÷ 11 = 251.5455 - This has a remainder. 11 is not a factor.
2,767 ÷ 13 = 212.8462 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,767 ÷ 2,767 = 1 - No remainder! 2,767 is one of the factors!

The orange divisor(s) above are the prime factors of the number 42,501,120. 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 3 x 5 x 2,767 = 42,501,120. It can also be written in exponential form as 210 x 31 x 51 x 2,7671.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 42,501,120.

42,501,120
Factor Arrows
221,250,560
Factor Arrows
210,625,280
Factor Arrows
25,312,640
Factor Arrows
22,656,320
Factor Arrows
21,328,160
Factor Arrows
2664,080
Factor Arrows
2332,040
Factor Arrows
2166,020
Factor Arrows
283,010
Factor Arrows
241,505
Factor Arrows
313,835
Factor Arrows
52,767

More Prime Factorization Examples

42,501,11842,501,11942,501,12142,501,122
21 x 111 x 2571 x 7,5171191 x 2,236,9011131 x 1,7091 x 1,913121 x 171 x 591 x 21,1871

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