Q: What is the prime factorization of the number 155,121,232?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 7 x 29 x 163 x 293
    • or also written as { 2, 2, 2, 2, 7, 29, 163, 293 }
  • Written in exponential form: 24 x 71 x 291 x 1631 x 2931

Why is the prime factorization of 155,121,232 written as 24 x 71 x 291 x 1631 x 2931?

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 155,121,232

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 155,121,232 by 2

155,121,232 ÷ 2 = 77,560,616 - No remainder! 2 is one of the factors!
77,560,616 ÷ 2 = 38,780,308 - No remainder! 2 is one of the factors!
38,780,308 ÷ 2 = 19,390,154 - No remainder! 2 is one of the factors!
19,390,154 ÷ 2 = 9,695,077 - No remainder! 2 is one of the factors!
9,695,077 ÷ 2 = 4,847,538.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
9,695,077 ÷ 3 = 3,231,692.3333 - This has a remainder. 3 is not a factor.
9,695,077 ÷ 5 = 1,939,015.4 - This has a remainder. 5 is not a factor.
9,695,077 ÷ 7 = 1,385,011 - No remainder! 7 is one of the factors!
1,385,011 ÷ 7 = 197,858.7143 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
1,385,011 ÷ 11 = 125,910.0909 - This has a remainder. 11 is not a factor.
1,385,011 ÷ 13 = 106,539.3077 - This has a remainder. 13 is not a factor.
1,385,011 ÷ 17 = 81,471.2353 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,385,011 ÷ 29 = 47,759 - No remainder! 29 is one of the factors!
47,759 ÷ 29 = 1,646.8621 - There is a remainder. We can't divide by 29 evenly anymore. Let's try the next prime number
47,759 ÷ 31 = 1,540.6129 - This has a remainder. 31 is not a factor.
47,759 ÷ 37 = 1,290.7838 - This has a remainder. 37 is not a factor.
47,759 ÷ 41 = 1,164.8537 - This has a remainder. 41 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
47,759 ÷ 163 = 293 - No remainder! 163 is one of the factors!
293 ÷ 163 = 1.7975 - There is a remainder. We can't divide by 163 evenly anymore. Let's try the next prime number
293 ÷ 167 = 1.7545 - This has a remainder. 167 is not a factor.
293 ÷ 173 = 1.6936 - This has a remainder. 173 is not a factor.
293 ÷ 179 = 1.6369 - This has a remainder. 179 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
293 ÷ 293 = 1 - No remainder! 293 is one of the factors!

The orange divisor(s) above are the prime factors of the number 155,121,232. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 7 x 29 x 163 x 293 = 155,121,232. It can also be written in exponential form as 24 x 71 x 291 x 1631 x 2931.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 155,121,232.

155,121,232
Factor Arrows
277,560,616
Factor Arrows
238,780,308
Factor Arrows
219,390,154
Factor Arrows
29,695,077
Factor Arrows
71,385,011
Factor Arrows
2947,759
Factor Arrows
163293

More Prime Factorization Examples

155,121,230155,121,231155,121,233155,121,234
21 x 51 x 111 x 9911 x 1,423131 x 611 x 847,65715991 x 258,967121 x 31 x 25,853,5391

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