C++ Coding Test Course, Finding the Least Common Multiple

1. Problem Description

This is a problem to find the Least Common Multiple (LCM) of two given integers A and B. The least common multiple refers to the smallest number among the multiples of the two numbers. For instance, the least common multiple of 4 and 5 is 20.

2. Input Format

In the first line, integers A and B are given. (1 ≤ A, B ≤ 106)

3. Output Format

The least common multiple of the given integers A and B will be output.

4. Problem Example

    Input:
    4 5

    Output:
    20

5. Algorithm Design

A common method for finding the least common multiple is to use the greatest common divisor of the two numbers. Here is the theory.

The least common multiple can be calculated as follows:

    LCM(A, B) = (A * B) / GCD(A, B)

Here, GCD refers to the Greatest Common Divisor. The Euclidean algorithm can be used to calculate it.

5.1 Euclidean Algorithm

The Euclidean algorithm is a classical method to find the GCD of two numbers, and it works as follows:

  1. If B is not 0, store the remainder of A divided by B in C.
  2. Update A to B and B to C.
  3. Repeat this process until B becomes 0.
  4. Finally, A will be the GCD.

6. C++ Code Implementation

Now, let’s implement the code to find the least common multiple in C++.


#include <iostream>
using namespace std;

// Greatest Common Divisor (GCD) calculation
int gcd(int a, int b) {
    while (b != 0) {
        int temp = b;
        b = a % b;
        a = temp;
    }
    return a;
}

// Least Common Multiple (LCM) calculation
int lcm(int a, int b) {
    return (a * b) / gcd(a, b);
}

int main() {
    int A, B;
    cout << "Please enter two integers A and B: ";
    cin >> A >> B;

    int result = lcm(A, B);
    cout << "The least common multiple is: " << result << endl;
    return 0;
}

7. Code Explanation

The above code can be divided into three main parts:

  1. gcd function: It calculates the greatest common divisor of two integers A and B.
  2. lcm function: It is responsible for calculating the least common multiple of two integers.
  3. main function: It serves as the entry point of the program, takes input from the user, and outputs the least common multiple.

8. Testing and Verification

To truly test the code, various input values should be used to verify the correct results.

    Input: 4 5
    Output: 20

    Input: 12 15
    Output: 60

    Input: 7 3
    Output: 21

    Input: 21 14
    Output: 42

    Input: 1 1000000
    Output: 1000000

Through these various test cases, the accuracy and reliability of the code can be verified.

9. Performance Considerations

The calculation of the least common multiple generally divides the product of the two numbers by the greatest common divisor, so the actual performance is greatly influenced by the performance of the GCD algorithm. The Euclidean algorithm has a time complexity of O(log(min(A, B))), which allows it to operate efficiently in proportion to the input size.

10. Conclusion

In this lesson, we learned how to find the least common multiple of two integers. We learned how to calculate the greatest common divisor using the Euclidean algorithm and how to use it to calculate the least common multiple. This algorithm can be applied to solve various problems, thus serving as a useful foundational knowledge.