Number of continuous reductions of A from B or B from A to make them (1, 1)

Given two integers A and B, the task is to find the minimum number of operations required to change this pair to (1, 1). In each operation (A, B) can be changed to (A – B, B) where A > B.

Note: If there is no possible solution to reach (1, 1), print -1.
Examples:

Input: A = 7, B = 8
Output: 7
Explanation:
Operation 1: (A, B) => (7, 8) => (7, 1)
Operation 2: (A, B) => (7, 1) => (6, 1)
Operation 3: (A, B) => (6, 1) => (5, 1)
Operation 4: (A, B) => (5, 1) => (4, 1)
Operation 5: (A, B) => (4, 1) => (3, 1)
Operation 6: (A, B) => (3, 1) => (2, 1)
Operation 7: (A, B) => (2, 1) => (1, 1)

Input: A = 75, B = 17
Output: 10

Naive Approach: The idea is to use recursion and update the pair as (A, A – B), where A > B and increase the number of operations required by 1. If at any step, any element of the pair is less than 1 then it is not possible to reach (1, 1).



Below is the implementation of the above approach:

C++

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// C++ implementation to find the
// minimum number of operations
// required to reach (1, 1)
#include <bits/stdc++.h>
using namespace std;
      
// Function to find the minimum
// number of steps required
int minimumSteps(int a, int b, int c)
{
      
    // Conditon to check if it 
    // is not possible to reach
    if(a < 1 || b < 1)
        return -1;
          
    // Condition to check if the 
    // pair is reached to 1, 1
    if(a == 1 && b == 1)
        return c;
      
    // Condition to change the 
    // A as the maximum element
    if(a < b)
    {
        a = a + b;
        b = a - b;
        a = a - b;
    }
      
    return minimumSteps(a - b, b, c + 1);
  
// Driver Code
int main()
{
    int a = 75;
    int b = 17;
      
    cout << minimumSteps(a, b, 0) << endl;
}
  
// This code is contributed by AbhiThakur

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Java

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// Java implementation to find the
// minimum number of operations
// required to reach (1, 1)
class GFG{
      
// Function to find the minimum
// number of steps required
static int minimumSteps(int a, int b, int c)
{
      
    // Conditon to check if it 
    // is not possible to reach
    if(a < 1 || b < 1)
        return -1;
          
    // Condition to check if the 
    // pair is reached to 1, 1
    if(a == 1 && b == 1)
        return c;
      
    // Condition to change the 
    // A as the maximum element
    if(a < b)
    {
        a = a + b;
        b = a - b;
        a = a - b;
    }
      
    return minimumSteps(a - b, b, c + 1);
  
// Driver Code
public static void main(String []args)
{
    int a = 75;
    int b = 17;
      
    System.out.println(minimumSteps(a, b, 0));
}
}
  
// This code is contributed by 29AjayKumar

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Python3

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# Python3 implementation to find the
# minimum number of operations
# required to reach (1, 1)
  
# Function to find the minimum
# number of steps required
def minimumSteps(a, b, c):
      
    # Conditon to check if it 
    # is not possible to reach
    if a < 1 or b < 1:
        return -1
          
    # Condition to check if the 
    # pair is reached to 1, 1
    if a == 1 and b == 1:
        return c
      
    # Condition to change the 
    # A as the maximum element
    if a < b:
        a, b = b, a
      
    return minimumSteps(a-b, b, c + 1)
      
# Driver Code
if __name__ == "__main__":
    a = 75; b = 17
      
    print(minimumSteps(a, b, 0))

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C#

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// C# implementation to find the
// minimum number of operations
// required to reach (1, 1)
using System;
  
class GFG{
      
// Function to find the minimum
// number of steps required
static int minimumSteps(int a, int b, int c)
{
      
    // Conditon to check if it 
    // is not possible to reach
    if(a < 1 || b < 1)
       return -1;
          
    // Condition to check if the 
    // pair is reached to 1, 1
    if(a == 1 && b == 1)
       return c;
      
    // Condition to change the 
    // A as the maximum element
    if(a < b)
    {
        a = a + b;
        b = a - b;
        a = a - b;
    }
    return minimumSteps(a - b, b, c + 1);
  
// Driver Code
public static void Main()
{
    int a = 75;
    int b = 17;
      
    Console.WriteLine(minimumSteps(a, b, 0));
}
}
  
// This code is contributed by AbhiThakur

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Output:

10

Effiecient Approach: The idea is to use Euclidean algorithm to solve this problem. By this approach, we can go from (A, B) to (A % B, B) in A/B steps. But if the minimum of the A and B is 1, then we can reach (1, 1) in A – 1 steps.

Below is the implementation of the above approach:

C++

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// C++ implementation to find the
// minimum number of operations
// required to reach (1, 1)
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the minimum
int minimumSteps(int a, int b, int c)
{
      
    // Conditon to check if it 
    // is not possible to reach
    if(a < 1 || b < 1)
    {
        return -1;
    }
      
    // Condition to check if the 
    // pair is reached to 1, 1
    if(min(a, b) == 1)
    {
        return c + max(a, b) - 1;
    }
      
    // Condition to change the 
    // A as the maximum element
    if(a < b)
    {
        swap(a, b);
    }
      
    return minimumSteps(a % b, b, c + (a / b));
}
  
// Driver Code
int main()
{
    int a = 75, b = 17;
    cout << minimumSteps(a, b, 0) << endl;
  
    return 0;
}
  
// This code is contributed by rutvik_56

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Java

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// Java implementation to find the
// minimum number of operations
// required to reach (1, 1)
import java.util.*;
class GFG{
  
// Function to find the minimum
static int minimumSteps(int a, int b, int c)
{
      
    // Conditon to check if it 
    // is not possible to reach
    if(a < 1 || b < 1)
    {
        return -1;
    }
      
    // Condition to check if the 
    // pair is reached to 1, 1
    if(Math.min(a, b) == 1)
    {
        return c + Math.max(a, b) - 1;
    }
      
    // Condition to change the 
    // A as the maximum element
    if(a < b)
    {
        a = a + b;
        b = a - b;
        a = a - b;
    }
      
    return minimumSteps(a % b, b, c + (a / b));
}
  
// Driver Code
public static void main(String[] args)
{
    int a = 75, b = 17;
    System.out.print(
           minimumSteps(a, b, 0) + "\n");
}
}
  
// This code is contributed by sapnasingh4991

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Python3

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# Python3 implementation to find the
# minimum number of operations
# required to reach (1, 1)
  
# Function to find the minimum
# number of steps required
def minimumSteps(a, b, c):
      
    # Conditon to check if it 
    # is not possible to reach
    if a < 1 or b < 1:
        return -1
      
    # Condition to check if the 
    # pair is reached to 1, 1
    if min(a, b) == 1:
        return c + max(a, b) - 1
          
    # Condition to change the 
    # A as the maximum element
    if a < b:
        a, b = b, a
          
    return minimumSteps(a % b, b, c + a//b)
      
# Driver Code
if __name__ == "__main__":
    a = 75; b = 17
      
    print(minimumSteps(a, b, 0))

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C#

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// C# implementation to find the
// minimum number of operations
// required to reach (1, 1)
using System;
class GFG{
  
// Function to find the minimum
static int minimumSteps(int a, int b, int c)
{
      
    // Conditon to check if it 
    // is not possible to reach
    if(a < 1 || b < 1)
    {
        return -1;
    }
      
    // Condition to check if the 
    // pair is reached to 1, 1
    if(Math.Min(a, b) == 1)
    {
        return c + Math.Max(a, b) - 1;
    }
      
    // Condition to change the 
    // A as the maximum element
    if(a < b)
    {
        a = a + b;
        b = a - b;
        a = a - b;
    }
      
    return minimumSteps(a % b, b, c + (a / b));
}
  
// Driver Code
public static void Main()
{
    int a = 75, b = 17;
    Console.Write(minimumSteps(a, b, 0) + "\n");
}
}
  
// This code is contributed by Nidhi_biet

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Output:

10

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