LCM (Least Common Multiple) and GCD (Greatest Common Divisor) are two important concepts in number theory. Here's a brief explanation of both:

• GCD: The largest positive integer that divides two or more integers without leaving a remainder.
• LCM: The smallest positive integer that is divisible by both of the given integers.

Relationship Between GCD and LCM

There is a relationship between GCD and LCM given by the formula:

$$\text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)}$$

This formula allows you to calculate LCM using GCD.

C Program to Calculate GCD and LCM

Here’s a C program that calculates both the GCD and LCM of two integers:

#include <stdio.h>

// Function to calculate GCD
int gcd(int a, int b) {
    if (b == 0)
        return a;
    return gcd(b, a % b); // Recursive call
}

// Function to calculate LCM using GCD
int lcm(int a, int b) {
    return (a * b) / gcd(a, b);
}

int main() {
    int num1, num2;

    // Input two numbers from the user
    printf("Enter two integers: ");
    scanf("%d %d", &num1, &num2);

    // Ensure that the input is positive
    if (num1 <= 0 || num2 <= 0) {
        printf("Please enter positive integers.\n");
        return 1; // Exit the program with an error code
    }

    // Calculate GCD and LCM
    int gcdResult = gcd(num1, num2);
    int lcmResult = lcm(num1, num2);

    // Output the results
    printf("GCD of %d and %d is: %d\n", num1, num2, gcdResult);
    printf("LCM of %d and %d is: %d\n", num1, num2, lcmResult);

    return 0;
}

Explanation:

1. Functions:

   • GCD Function:
     • Uses the Euclidean algorithm to calculate the GCD.
     • If b is zero, a is the GCD. Otherwise, it calls itself recursively with b and a % b.
   • LCM Function:
     • Uses the formula mentioned earlier to calculate LCM based on the GCD.

2. Input:

   • The program prompts the user to enter two integers.

3. Validation:

   • It checks whether the input numbers are positive. If not, it prints a message and exits the program.

4. Calculations:

   • The program calculates the GCD using the gcd function and then calculates the LCM using the lcm function.

5. Output:

   • Finally, it prints the GCD and LCM of the two numbers.

Sample Output:

Example 1:

Enter two integers: 12 15
GCD of 12 and 15 is: 3
LCM of 12 and 15 is: 60

Example 2:

Enter two integers: 18 24
GCD of 18 and 24 is: 6
LCM of 18 and 24 is: 72

Example 3:

Enter two integers: 10 0
Please enter positive integers.

Key Points:

• GCD Calculation: The GCD function is efficient and leverages recursion.
• LCM Calculation: The relationship between GCD and LCM simplifies the LCM calculation.
• Input Validation: The program includes a simple validation step to ensure positive input.
• Modularity: The separation of GCD and LCM into different functions enhances code readability and reusability.