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Minimize steps to make given two number equal by adding LSB

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  • Last Updated : 29 Aug, 2022
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Given two even numbers A, and B, the task is to find the minimum number of operations required to make both numbers equal if it is possible else print -1. In a single operation, the last digit of either number A or B can be added to itself i.e., 

  • Either A = (A + (A % 10)) 
  • or B = (B + (B % 10))

Examples: 

Input: A = 4, B = 12 
Output: -1 
Explanation: We can not make 4 and 12 equal so answer is -1.

Input: A = 2 , B  = 8 
Output: 2. 
Explanation: 2 -> 4 -> 8, so minimum 2 steps is required to make them equal.

Input: A = 6, B = 12 .
Output: 1.
Explanation: 6 + 6 = 12 and 12 so by 1 step we can make elements equal.

Approach: To solve the problem follow the below observation:

Observation:

  • There is an observation that if last digit is 0 then we can not increase that number . So if last digit of the either number is 0 then the numbers should be equal else answer is -1 .
  • The last digit of the even numbers contain 2, 4, 6, 8 . We will see the change if last digit is added  2->4->8->6->2 ->4->8->6->2 . . . 
  • So the pattern is repeating and in every cycle the increase in the number is 2+4+8+6=20 .
  • Now if two number will be equal then last digit of the numbers must be equal .
  • Now we can make last digit of the two numbers either equal to 2 or 4 or 6 or 8 and check if the difference between two numbers modulo 20 is equal to 0 or not .
  • If modulo is 0 then possible to make them equal else not because if difference%20 = 0 then we can increase the smaller number by any multiple of 20 to make it equal to greater even number .
  • To find the minimum number of steps we will make last digit of both numbers equal to the last digit of the greater number and find the number of steps to make them equal .

Follow the steps to solve the problem:

  • If last digit of any number is 0.
    • if the numbers are the same then the answer is 0.
    • else the number is -1 as the numbers can’t be made equal.
  • Store the minimum and maximum of A and B in different variables.
  • Make the last digit of the smaller number equal to the greater number and count the steps.
  • If the difference is not a multiple of 20 return -1 as the answer.
  • Calculate the steps in the loop by using formula (((c – d) / 20) * 4).
  • Return the sum of steps and loopSteps as the final answer.

Below is the implementation of the above approach : 

C++14




// C++ code for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to count minimum steps
int ispossible(int A, int B)
{
 
    // If any number has 0 as last Digit
    if (A % 10 == 0 || B % 10 == 0) {
 
        // If the number are not equal
        // they can never be made equal
        if (A != B) {
            return -1;
        }
        else {
            return 0;
        }
    }
 
    // Storing the minimum of A and B
    int c = min(A, B);
 
    // Storing the maximum of A and B
    int d = max(A, B);
 
    // Steps to make the last digit equal
    int steps = 0;
 
    // To make the last digit same
    int lastDigit = d % 10;
 
    while (c % 10 != lastDigit) {
        c += (c % 10);
        steps++;
    }
 
    if ((d - c) % 20 != 0) {
 
        // If cycle is not formed
        // after same last digit
        return -1;
    }
 
    // Steps in the loop
    int loopSteps = (((d - c) / 20) * 4);
 
    int ans = steps + loopSteps;
    return ans;
}
 
// Driver Code
int main()
{
 
    // Given Inputs
    int A = 2, B = 8;
 
    // Function Call
    cout << ispossible(A, B);
    return 0;
}

Java




// Java code for the above approach
import java.io.*;
 
class GFG {
 
  // Driver Code
  public static void main(String[] args) {
    // Given Inputs
    int A = 2, B = 8;
    // Function Call
    System.out.println(ispossible(A, B));
  }
 
  // Function to count minimum steps
  static int ispossible(int A, int B) {
 
    // If any number has 0 as last Digit
    if (A % 10 == 0 || B % 10 == 0) {
 
      // If the number are not equal
      // they can never be made equal
      if (A != B) {
        return -1;
      } else {
        return 0;
      }
    }
 
    // Storing the minimum of A and B
    int c = min(A, B);
 
    // Storing the maximum of A and B
    int d = max(A, B);
 
    // Steps to make the last digit equal
    int steps = 0;
 
    // To make the last digit same
    int lastDigit = d % 10;
 
    while (c % 10 != lastDigit) {
      c += (c % 10);
      steps++;
    }
 
    if ((d - c) % 20 != 0) {
 
      // If cycle is not formed
      // after same last digit
      return -1;
    }
 
    // Steps in the loop
    int loopSteps = (((d - c) / 20) * 4);
 
    int ans = steps + loopSteps;
    return ans;
  }
 
  // function to find maximum element 
  static int max(int a, int b) {
    if (a > b) return a;
    return b;
  }
 
  // function to find minimum element
  static int min(int a, int b) {
    if (a < b) return a;
    return b;
  }
}
 
// This code is contributed by ajaymakvana.

Python3




# Python3 code for the above approach
 
# Function to count minimum steps
def ispossible(A, B) :
 
    # If any number has 0 as last Digit
    if (A % 10 == 0 or B % 10 == 0) :
 
        # If the number are not equal
        # they can never be made equal
        if (A != B) :
            return -1;
         
        else :
            return 0;
 
    # Storing the minimum of A and B
    c = min(A, B);
 
    # Storing the maximum of A and B
    d = max(A, B);
 
    # Steps to make the last digit equal
    steps = 0;
 
    # To make the last digit same
    lastDigit = d % 10;
 
    while (c % 10 != lastDigit) :
        c += (c % 10);
        steps += 1;
 
    if ((d - c) % 20 != 0) :
 
        # If cycle is not formed
        # after same last digit
        return -1;
 
    # Steps in the loop
    loopSteps = (((d - c) / 20) * 4);
 
    ans = steps + loopSteps;
    return ans;
 
# Driver Code
if __name__ == "__main__" :
 
    # Given Inputs
    A = 2; B = 8;
 
    # Function Call
    print(ispossible(A, B));
    
   # This code is contributed by AnkThon

C#




// C# program for above approach
using System;
class GFG
{
 
// Function to count minimum steps
  static int ispossible(int A, int B) {
 
    // If any number has 0 as last Digit
    if (A % 10 == 0 || B % 10 == 0) {
 
      // If the number are not equal
      // they can never be made equal
      if (A != B) {
        return -1;
      } else {
        return 0;
      }
    }
 
    // Storing the minimum of A and B
    int c = min(A, B);
 
    // Storing the maximum of A and B
    int d = max(A, B);
 
    // Steps to make the last digit equal
    int steps = 0;
 
    // To make the last digit same
    int lastDigit = d % 10;
 
    while (c % 10 != lastDigit) {
      c += (c % 10);
      steps++;
    }
 
    if ((d - c) % 20 != 0) {
 
      // If cycle is not formed
      // after same last digit
      return -1;
    }
 
    // Steps in the loop
    int loopSteps = (((d - c) / 20) * 4);
 
    int ans = steps + loopSteps;
    return ans;
  }
 
  // function to find maximum element 
  static int max(int a, int b) {
    if (a > b) return a;
    return b;
  }
 
  // function to find minimum element
  static int min(int a, int b) {
    if (a < b) return a;
    return b;
  }
 
 
// Driver Code
public static void Main()
{
    // Given Inputs
    int A = 2, B = 8;
    // Function Call
    Console.Write(ispossible(A, B));
}
}
 
// This code is contributed by code_hunt.

Javascript




<script>
// JavaScript code for the above approach
 
// Function to count minimum steps
  function ispossible(A, B) {
 
    // If any number has 0 as last Digit
    if (A % 10 == 0 || B % 10 == 0) {
 
      // If the number are not equal
      // they can never be made equal
      if (A != B) {
        return -1;
      } else {
        return 0;
      }
    }
 
    // Storing the minimum of A and B
    let c = min(A, B);
 
    // Storing the maximum of A and B
    let d = max(A, B);
 
    // Steps to make the last digit equal
    let steps = 0;
 
    // To make the last digit same
    let lastDigit = d % 10;
 
    while (c % 10 != lastDigit) {
      c += (c % 10);
      steps++;
    }
 
    if ((d - c) % 20 != 0) {
 
      // If cycle is not formed
      // after same last digit
      return -1;
    }
 
    // Steps in the loop
    let loopSteps = (((d - c) / 20) * 4);
 
    let ans = steps + loopSteps;
    return ans;
  }
 
  // function to find maximum element 
  function max(a, b) {
    if (a > b) return a;
    return b;
  }
 
  // function to find minimum element
  function min(a, b) {
    if (a < b) return a;
    return b;
  }
 
// Driver Code
     
    // Given Inputs
    let A = 2, B = 8;
    // Function Call
    document.write(ispossible(A, B));
     
    // This code is contributed by sanjoy_62.
</script>

Output

2

Time complexity: O(1), as we will perform at most 3 steps in the while loop. 
Auxiliary Space: O(1), since we did not use any extra space.


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