Q: What is the prime factorization of the number 141,241,120?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 5 x 197 x 4,481
    • or also written as { 2, 2, 2, 2, 2, 5, 197, 4,481 }
  • Written in exponential form: 25 x 51 x 1971 x 4,4811

Why is the prime factorization of 141,241,120 written as 25 x 51 x 1971 x 4,4811?

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 141,241,120

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 141,241,120 by 2

141,241,120 ÷ 2 = 70,620,560 - No remainder! 2 is one of the factors!
70,620,560 ÷ 2 = 35,310,280 - No remainder! 2 is one of the factors!
35,310,280 ÷ 2 = 17,655,140 - No remainder! 2 is one of the factors!
17,655,140 ÷ 2 = 8,827,570 - No remainder! 2 is one of the factors!
8,827,570 ÷ 2 = 4,413,785 - No remainder! 2 is one of the factors!
4,413,785 ÷ 2 = 2,206,892.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
4,413,785 ÷ 3 = 1,471,261.6667 - This has a remainder. 3 is not a factor.
4,413,785 ÷ 5 = 882,757 - No remainder! 5 is one of the factors!
882,757 ÷ 5 = 176,551.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
882,757 ÷ 7 = 126,108.1429 - This has a remainder. 7 is not a factor.
882,757 ÷ 11 = 80,250.6364 - This has a remainder. 11 is not a factor.
882,757 ÷ 13 = 67,904.3846 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
882,757 ÷ 197 = 4,481 - No remainder! 197 is one of the factors!
4,481 ÷ 197 = 22.7462 - There is a remainder. We can't divide by 197 evenly anymore. Let's try the next prime number
4,481 ÷ 199 = 22.5176 - This has a remainder. 199 is not a factor.
4,481 ÷ 211 = 21.237 - This has a remainder. 211 is not a factor.
4,481 ÷ 223 = 20.0942 - This has a remainder. 223 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
4,481 ÷ 4,481 = 1 - No remainder! 4,481 is one of the factors!

The orange divisor(s) above are the prime factors of the number 141,241,120. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 5 x 197 x 4,481 = 141,241,120. It can also be written in exponential form as 25 x 51 x 1971 x 4,4811.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 141,241,120.

141,241,120
Factor Arrows
270,620,560
Factor Arrows
235,310,280
Factor Arrows
217,655,140
Factor Arrows
28,827,570
Factor Arrows
24,413,785
Factor Arrows
5882,757
Factor Arrows
1974,481

More Prime Factorization Examples

141,241,118141,241,119141,241,121141,241,122
21 x 971 x 728,047131 x 1031 x 457,091171 x 20,177,303121 x 32 x 112 x 64,8491

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