Q: What is the prime factorization of the number 21,035,007?

 A:
  • The prime factors are: 3 x 3 x 7 x 233 x 1,433
    • or also written as { 3, 3, 7, 233, 1,433 }
  • Written in exponential form: 32 x 71 x 2331 x 1,4331

Why is the prime factorization of 21,035,007 written as 32 x 71 x 2331 x 1,4331?

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 21,035,007

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 21,035,007 by 2

21,035,007 ÷ 2 = 10,517,503.5 - This has a remainder. Let's try another prime number.
21,035,007 ÷ 3 = 7,011,669 - No remainder! 3 is one of the factors!
7,011,669 ÷ 3 = 2,337,223 - No remainder! 3 is one of the factors!
2,337,223 ÷ 3 = 779,074.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
2,337,223 ÷ 5 = 467,444.6 - This has a remainder. 5 is not a factor.
2,337,223 ÷ 7 = 333,889 - No remainder! 7 is one of the factors!
333,889 ÷ 7 = 47,698.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
333,889 ÷ 11 = 30,353.5455 - This has a remainder. 11 is not a factor.
333,889 ÷ 13 = 25,683.7692 - This has a remainder. 13 is not a factor.
333,889 ÷ 17 = 19,640.5294 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
333,889 ÷ 233 = 1,433 - No remainder! 233 is one of the factors!
1,433 ÷ 233 = 6.1502 - There is a remainder. We can't divide by 233 evenly anymore. Let's try the next prime number
1,433 ÷ 239 = 5.9958 - This has a remainder. 239 is not a factor.
1,433 ÷ 241 = 5.9461 - This has a remainder. 241 is not a factor.
1,433 ÷ 251 = 5.7092 - This has a remainder. 251 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,433 ÷ 1,433 = 1 - No remainder! 1,433 is one of the factors!

The orange divisor(s) above are the prime factors of the number 21,035,007. If we put all of it together we have the factors 3 x 3 x 7 x 233 x 1,433 = 21,035,007. It can also be written in exponential form as 32 x 71 x 2331 x 1,4331.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 21,035,007.

21,035,007
Factor Arrows
37,011,669
Factor Arrows
32,337,223
Factor Arrows
7333,889
Factor Arrows
2331,433

More Prime Factorization Examples

21,035,00521,035,00621,035,00821,035,009
51 x 291 x 145,069121 x 1791 x 58,7571211 x 10,2711411 x 1391 x 3,6911

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