Q: What is the prime factorization of the number 22,161,246?

 A:
  • The prime factors are: 2 x 3 x 127 x 127 x 229
    • or also written as { 2, 3, 127, 127, 229 }
  • Written in exponential form: 21 x 31 x 1272 x 2291

Why is the prime factorization of 22,161,246 written as 21 x 31 x 1272 x 2291?

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 22,161,246

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 22,161,246 by 2

22,161,246 ÷ 2 = 11,080,623 - No remainder! 2 is one of the factors!
11,080,623 ÷ 2 = 5,540,311.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
11,080,623 ÷ 3 = 3,693,541 - No remainder! 3 is one of the factors!
3,693,541 ÷ 3 = 1,231,180.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
3,693,541 ÷ 5 = 738,708.2 - This has a remainder. 5 is not a factor.
3,693,541 ÷ 7 = 527,648.7143 - This has a remainder. 7 is not a factor.
3,693,541 ÷ 11 = 335,776.4545 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
3,693,541 ÷ 127 = 29,083 - No remainder! 127 is one of the factors!
29,083 ÷ 127 = 229 - No remainder! 127 is one of the factors!
229 ÷ 127 = 1.8031 - There is a remainder. We can't divide by 127 evenly anymore. Let's try the next prime number
229 ÷ 131 = 1.7481 - This has a remainder. 131 is not a factor.
229 ÷ 137 = 1.6715 - This has a remainder. 137 is not a factor.
229 ÷ 139 = 1.6475 - This has a remainder. 139 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
229 ÷ 229 = 1 - No remainder! 229 is one of the factors!

The orange divisor(s) above are the prime factors of the number 22,161,246. If we put all of it together we have the factors 2 x 3 x 127 x 127 x 229 = 22,161,246. It can also be written in exponential form as 21 x 31 x 1272 x 2291.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 22,161,246.

22,161,246
Factor Arrows
211,080,623
Factor Arrows
33,693,541
Factor Arrows
12729,083
Factor Arrows
127229

More Prime Factorization Examples

22,161,24422,161,24522,161,24722,161,248
22 x 71 x 791,473151 x 4,432,24911,9071 x 11,621125 x 692,5391

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