Q: What is the prime factorization of the number 144,204,234?

 A:
  • The prime factors are: 2 x 3 x 17 x 67 x 21,101
    • or also written as { 2, 3, 17, 67, 21,101 }
  • Written in exponential form: 21 x 31 x 171 x 671 x 21,1011

Why is the prime factorization of 144,204,234 written as 21 x 31 x 171 x 671 x 21,1011?

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 144,204,234

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 144,204,234 by 2

144,204,234 ÷ 2 = 72,102,117 - No remainder! 2 is one of the factors!
72,102,117 ÷ 2 = 36,051,058.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
72,102,117 ÷ 3 = 24,034,039 - No remainder! 3 is one of the factors!
24,034,039 ÷ 3 = 8,011,346.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
24,034,039 ÷ 5 = 4,806,807.8 - This has a remainder. 5 is not a factor.
24,034,039 ÷ 7 = 3,433,434.1429 - This has a remainder. 7 is not a factor.
24,034,039 ÷ 11 = 2,184,912.6364 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
24,034,039 ÷ 17 = 1,413,767 - No remainder! 17 is one of the factors!
1,413,767 ÷ 17 = 83,162.7647 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
1,413,767 ÷ 19 = 74,408.7895 - This has a remainder. 19 is not a factor.
1,413,767 ÷ 23 = 61,468.1304 - This has a remainder. 23 is not a factor.
1,413,767 ÷ 29 = 48,750.5862 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,413,767 ÷ 67 = 21,101 - No remainder! 67 is one of the factors!
21,101 ÷ 67 = 314.9403 - There is a remainder. We can't divide by 67 evenly anymore. Let's try the next prime number
21,101 ÷ 71 = 297.1972 - This has a remainder. 71 is not a factor.
21,101 ÷ 73 = 289.0548 - This has a remainder. 73 is not a factor.
21,101 ÷ 79 = 267.1013 - This has a remainder. 79 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
21,101 ÷ 21,101 = 1 - No remainder! 21,101 is one of the factors!

The orange divisor(s) above are the prime factors of the number 144,204,234. If we put all of it together we have the factors 2 x 3 x 17 x 67 x 21,101 = 144,204,234. It can also be written in exponential form as 21 x 31 x 171 x 671 x 21,1011.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 144,204,234.

144,204,234
Factor Arrows
272,102,117
Factor Arrows
324,034,039
Factor Arrows
171,413,767
Factor Arrows
6721,101

More Prime Factorization Examples

144,204,232144,204,233144,204,235144,204,236
23 x 3,2591 x 5,53118771 x 164,429151 x 71 x 2391 x 17,239122 x 111 x 1911 x 17,1591

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