Orig Description
Score : 300 points
Problem StatementFind \displaystyle{\sum_{a=1}^{K}\sum_{b=1}^{K}\sum_{c=1}^{K} \gcd(a,b,c)}.
Here \gcd(a,b,c) denotes the greatest common divisor of a, b, and c.
Constraints
1 \leq K \leq 200
K is an integer.
InputInput is given from Standard Input in the following format:
K
OutputPrint the value of \displaystyle{\sum_{a=1}^{K}\sum_{b=1}^{K}\sum_{c=1}^{K} \gcd(a,b,c)}.
Sample Input 12
Sample Output 19
\gcd(1,1,1)+\gcd(1,1,2)+\gcd(1,2,1)+\gcd(1,2,2)
+\gcd(2,1,1)+\gcd(2,1,2)+\gcd(2,2,1)+\gcd(2,2,2)
=1+1+1+1+1+1+1+2=9
Thus, the answer is 9.
Sample Input 2200
Sample Output 210813692
Test Case Generator
Certainly! Below is the test case generator for the given problem statement:
```python
import random
def tcgen():
# Generate a random integer K such that 1 ≤ K ≤ 200
K = random.randint(1, 200)
return K
# Example usage
if __name__ == "__main__":
# Generate a test case
test_case = tcgen()
# Print the test case in the required format
print(test_case)
```
This generator produces a random integer \( K \) within the specified range, which is the input for the problem. Each time you run the generator, it will produce a different value of \( K \).
For a complete testing setup, you might also want to include the function to calculate the output based on the generated \( K \). Here is an example implementation for the calculation:
```python
import math
def calculate_gcd_sum(K):
gcd_sum = 0
for a in range(1, K + 1):
for b in range(1, K + 1):
for c in range(1, K + 1):
gcd_sum += math.gcd(math.gcd(a, b), c)
return gcd_sum
# Example usage
if __name__ == "__main__":
# Generate a test case
K = tcgen()
# Print the test case in the required format
print(f"Input: {K}")
# Calculate the output
result = calculate_gcd_sum(K)
# Print the result
print(f"Output: {result}")
```
This code will generate a random test case, print it, and then calculate and print the expected output based on the problem's requirements. The `calculate_gcd_sum` function computes the sum of the greatest common divisors for all triplets \((a, b, c)\) where \(1 \leq a, b, c \leq K\).