GCD, LCM and Distributive Property

3.7

Given three integers x, y, z, the task is to compute the value of GCD(LCM(x,y), LCM(x,z)).
Where, GCD = Greatest Common Divisor, LCM = Least Common Multiple

Examples:

Input: x = 15, y = 20, z = 100
Output: 60

Input: x = 30, y = 40, z = 400
Output: 120

One way to solve it is by finding GCD(x, y), and using it we find LCM(x, y). Similarly, we find LCM(x, z) and then we finally find the GCD of the obtained results.

An efficient approach can be done by the fact that the following version of distributivity holds true:

GCD(LCM (x, y), LCM (x, z)) = LCM(x, GCD(y, z))

For example, GCD(LCM(3, 4), LCM(3, 10)) = LCM(3, GCD(4, 10)) = LCM(3, 20) = 60

This reduces our work to compute the given problem statement.

// C++ program to compute value of GCD(LCM(x,y), LCM(x,z))
#include<bits/stdc++.h>
using namespace std;

// Returns value of  GCD(LCM(x,y), LCM(x,z))
int findValue(int x, int y, int z)
{
    int g = __gcd(y, z);

    // Return LCM(x, GCD(y, z))
    return (x*g)/__gcd(x, g);
}

int main()
{
    int x = 30, y = 40, z = 400;
    cout << findValue(x, y, z);
    return 0;
}

Output:

120

As a side note, vice versa is also true, i.e., gcd(x, lcm(y, z)) = lcm(gcd(x, y), gcd(x, z)

Reference:
https://en.wikipedia.org/wiki/Distributive_property#Other_examples

This article is contributed by Mazhar Imam Khan. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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