Q: What is the prime factorization of the number 203,800,576?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 7 x 1,777
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 1,777 }
  • Written in exponential form: 214 x 71 x 1,7771

Why is the prime factorization of 203,800,576 written as 214 x 71 x 1,7771?

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 203,800,576

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 203,800,576 by 2

203,800,576 ÷ 2 = 101,900,288 - No remainder! 2 is one of the factors!
101,900,288 ÷ 2 = 50,950,144 - No remainder! 2 is one of the factors!
50,950,144 ÷ 2 = 25,475,072 - No remainder! 2 is one of the factors!
25,475,072 ÷ 2 = 12,737,536 - No remainder! 2 is one of the factors!
12,737,536 ÷ 2 = 6,368,768 - No remainder! 2 is one of the factors!
6,368,768 ÷ 2 = 3,184,384 - No remainder! 2 is one of the factors!
3,184,384 ÷ 2 = 1,592,192 - No remainder! 2 is one of the factors!
1,592,192 ÷ 2 = 796,096 - No remainder! 2 is one of the factors!
796,096 ÷ 2 = 398,048 - No remainder! 2 is one of the factors!
398,048 ÷ 2 = 199,024 - No remainder! 2 is one of the factors!
199,024 ÷ 2 = 99,512 - No remainder! 2 is one of the factors!
99,512 ÷ 2 = 49,756 - No remainder! 2 is one of the factors!
49,756 ÷ 2 = 24,878 - No remainder! 2 is one of the factors!
24,878 ÷ 2 = 12,439 - No remainder! 2 is one of the factors!
12,439 ÷ 2 = 6,219.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
12,439 ÷ 3 = 4,146.3333 - This has a remainder. 3 is not a factor.
12,439 ÷ 5 = 2,487.8 - This has a remainder. 5 is not a factor.
12,439 ÷ 7 = 1,777 - No remainder! 7 is one of the factors!
1,777 ÷ 7 = 253.8571 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
1,777 ÷ 11 = 161.5455 - This has a remainder. 11 is not a factor.
1,777 ÷ 13 = 136.6923 - This has a remainder. 13 is not a factor.
1,777 ÷ 17 = 104.5294 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,777 ÷ 1,777 = 1 - No remainder! 1,777 is one of the factors!

The orange divisor(s) above are the prime factors of the number 203,800,576. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 7 x 1,777 = 203,800,576. It can also be written in exponential form as 214 x 71 x 1,7771.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 203,800,576.

203,800,576
Factor Arrows
2101,900,288
Factor Arrows
250,950,144
Factor Arrows
225,475,072
Factor Arrows
212,737,536
Factor Arrows
26,368,768
Factor Arrows
23,184,384
Factor Arrows
21,592,192
Factor Arrows
2796,096
Factor Arrows
2398,048
Factor Arrows
2199,024
Factor Arrows
299,512
Factor Arrows
249,756
Factor Arrows
224,878
Factor Arrows
212,439
Factor Arrows
71,777

More Prime Factorization Examples

203,800,574203,800,575203,800,577203,800,578
21 x 191 x 291 x 1731 x 1,069131 x 52 x 111 x 247,0311203,800,577121 x 31 x 1,4091 x 24,1071

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