GCD of two numbers formed by n repeating x and y times

3.7

Given three positive integer n, x, y. The task is to print Greatest Common Divisor of numbers formed by n repeating x times and number formed by n repeating y times.
0 <= n, x, y <= 1000000000.

Examples:

Input : n = 123, x = 2, y = 3.
Output : 123
Number formed are 123123 and 123123123.
Greatest Common Divisor of 123123 and
123123123 is 123.

Input : n = 4, x = 4, y = 6.
Output : 44

The idea is based on Euclidean algorithm to compute GCD of two number.
Let f(n, x) be a function which gives a number n repeated x times. So, we need to find GCD(f(n, x), f(n, y)).

Let n = 123, x = 3, y = 2.
So, first number A is f(123, 3) = 123123123 and second number B is f(123, 2) = 123123. We know, GCD(A,B) = GCD(A – B, B), using this property we can substract any multiple of B, say B’ from first A as long as B’ is smaller than A.
So, A = 123123123 and B’ can be 123123000. On substracting A will became 123 and B remains same.
Therfore, A = A – B’ = f(n, x – y).
So, GCD(f(n, x), f(n, y)) = GCD(f(n, x – y), f(n, y))

We can conclude following,

GCD(f(n, x), f(n, y)) = f(n, GCD(x, y)). 

Below is C++ implementation based on this approach:

// C++ program to print Greatest Common Divisor
// of number formed by N repeating x times and
// y times.
#include<bits/stdc++.h>
using namespace std;

// Return the Greatest common Divisor of two numbers.
int gcd(int a, int b)
{
    if (a == 0)
        return b;
    return gcd(b%a, a);
}

// Prints Greatest Common Divisor of number formed
// by n repeating x times and y times.
void findgcd(int n, int x, int y)
{
    // Finding GCD of x and y.
    int g = gcd(x,y);

    // Print n, g times.
    for (int i = 0; i < g; i++)
        cout << n;
}

// Driven Program
int main()
{
    int n = 123, x = 5, y = 2;
    findgcd(n, x, y);
    return 0;
}

Output:

123

This article is contributed by Anuj Chauhan. 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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