Q: What is the prime factorization of the number 101,024,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5 x 7 x 11 x 41
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 5, 7, 11, 41 }
  • Written in exponential form: 28 x 53 x 71 x 111 x 411

Why is the prime factorization of 101,024,000 written as 28 x 53 x 71 x 111 x 411?

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 101,024,000

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 101,024,000 by 2

101,024,000 ÷ 2 = 50,512,000 - No remainder! 2 is one of the factors!
50,512,000 ÷ 2 = 25,256,000 - No remainder! 2 is one of the factors!
25,256,000 ÷ 2 = 12,628,000 - No remainder! 2 is one of the factors!
12,628,000 ÷ 2 = 6,314,000 - No remainder! 2 is one of the factors!
6,314,000 ÷ 2 = 3,157,000 - No remainder! 2 is one of the factors!
3,157,000 ÷ 2 = 1,578,500 - No remainder! 2 is one of the factors!
1,578,500 ÷ 2 = 789,250 - No remainder! 2 is one of the factors!
789,250 ÷ 2 = 394,625 - No remainder! 2 is one of the factors!
394,625 ÷ 2 = 197,312.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
394,625 ÷ 3 = 131,541.6667 - This has a remainder. 3 is not a factor.
394,625 ÷ 5 = 78,925 - No remainder! 5 is one of the factors!
78,925 ÷ 5 = 15,785 - No remainder! 5 is one of the factors!
15,785 ÷ 5 = 3,157 - No remainder! 5 is one of the factors!
3,157 ÷ 5 = 631.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
3,157 ÷ 7 = 451 - No remainder! 7 is one of the factors!
451 ÷ 7 = 64.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
451 ÷ 11 = 41 - No remainder! 11 is one of the factors!
41 ÷ 11 = 3.7273 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
41 ÷ 13 = 3.1538 - This has a remainder. 13 is not a factor.
41 ÷ 17 = 2.4118 - This has a remainder. 17 is not a factor.
41 ÷ 19 = 2.1579 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
41 ÷ 41 = 1 - No remainder! 41 is one of the factors!

The orange divisor(s) above are the prime factors of the number 101,024,000. 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 5 x 5 x 5 x 7 x 11 x 41 = 101,024,000. It can also be written in exponential form as 28 x 53 x 71 x 111 x 411.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 101,024,000.

101,024,000
Factor Arrows
250,512,000
Factor Arrows
225,256,000
Factor Arrows
212,628,000
Factor Arrows
26,314,000
Factor Arrows
23,157,000
Factor Arrows
21,578,500
Factor Arrows
2789,250
Factor Arrows
2394,625
Factor Arrows
578,925
Factor Arrows
515,785
Factor Arrows
53,157
Factor Arrows
7451
Factor Arrows
1141

More Prime Factorization Examples

101,023,998101,023,999101,024,001101,024,002
21 x 31 x 16,837,3331101,023,999132 x 131 x 4611 x 1,873121 x 4211 x 119,9811

Try the factor calculator

Explore more about the number 101,024,000:


Ask a Question