Check if X can be reduced to 0 in exactly T moves by subtracting D or 1 from it

Given an integer X, D and T, the task is to check if it is possible to reduce X to 0 in exactly T moves. In each move, X can be reduced by either D or 1. Print YES if possible, else NO.

Example:

Input: X = 10, D = 3, T = 6
Output: YES
Explanation: Below are the moves applied- 
X = X – 3 = 7
X = X – 3 = 4
X = X – 1 = 3
X = X – 1 = 2
X = X – 1 = 1
X = X – 1 = 0
Hence it is possible to make X = 0 after exactly T moves.

Input: X = 10, D = 3, T = 5
Output: NO

Naive Approach: Check for possible moves of reducing D and 1, and check if X can be made to 0 in exactly T moves.
Time Complexity: O(X)

Efficient Approach:Given problem can be solved using following steps:

• Let K be the number of times X is reduced by D
• So the total distance will be K*(D) + (T – K), and it should be equal to X. 
  Therefore the equation becomes 
  

=> K*(D – 1) + T = X 
=> K*(D – 1) = X – T 

• For a valid answer to exist, (X – T) should be divisible by (D – 1)

Below is the implementation of the above approach:

C++

// C++ Program to implement the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to check the above problem condition
int possibleReachingSequence(int X, int D, int T)
{
    // Check for base cases
    if (X < T) {
        cout << "NO";
        return 0;
    }
    if (T * D < X) {
        cout << "NO";
        return 0;
    }
    // Check if D - 1 is a divisor of X - T
    if ((X - T) % (D - 1) == 0) {
        cout << "YES";
    }
    else {
        cout << "NO";
    }
    return 0;
}

// Driver Code
int main()
{
    int X = 10, D = 3, T = 6;
    possibleReachingSequence(X, D, T);
}

Java

// Java program for the above approach
import java.util.*;

class GFG{

// Function to check the above problem condition
static int possibleReachingSequence(int X, int D, int T)
{
    // Check for base cases
    if (X < T) {
        System.out.println("NO");
        return 0;
    }
    if (T * D < X) {
        System.out.println("NO");
        return 0;
    }
    // Check if D - 1 is a divisor of X - T
    if ((X - T) % (D - 1) == 0) {
        System.out.println("YES");
    }
    else {
        System.out.println("NO");
    }
    return 0;
}


// Driver Code
public static void main(String args[])
{
    int X = 10, D = 3, T = 6;
    possibleReachingSequence(X, D, T);
}
}

Python3

# Python program for the above approach

# Function to check the above problem condition
def possibleReachingSequence(X, D, T):

  # Check the base case.
  if X < T:
    return "NO"
  if T*D < X:
    return "NO"

  # check if X-T is a divisor of D-1
  if (X - T) % (D - 1) == 0:
    return "YES"
  return "NO"

# Driver code
X = 10
D = 3
T = 6
print(possibleReachingSequence(X, D, T))

# This code is contributed by maddler.

C#

// C# program for the above approach

using System;

public class GFG{

// Function to check the above problem condition
static int possibleReachingSequence(int X, int D, int T)
{
    // Check for base cases
    if (X < T) {
        Console.WriteLine("NO");
        return 0;
    }
    if (T * D < X) {
        Console.WriteLine("NO");
        return 0;
    }
    // Check if D - 1 is a divisor of X - T
    if ((X - T) % (D - 1) == 0) {
        Console.WriteLine("YES");
    }
    else {
        Console.WriteLine("NO");
    }
    return 0;
}


// Driver Code
public static void Main(string []args)
{
    int X = 10, D = 3, T = 6;
    possibleReachingSequence(X, D, T);
}
}

// This code is contributed by AnkThon

Javascript

<script>
// Javascript Program to implement the above approach

// Function to check the above problem condition
function possibleReachingSequence(X, D, T)
{

  // Check for base cases
  if (X < T) {
    document.write("NO");
    return 0;
  }
  if (T * D < X) {
    document.write("NO");
    return 0;
  }

  // Check if D - 1 is a divisor of X - T
  if ((X - T) % (D - 1) == 0) {
    document.write("YES");
  } else {
    document.write("NO");
  }
  return 0;
}

// Driver Code

let X = 10,
  D = 3,
  T = 6;
possibleReachingSequence(X, D, T);

// This code is contributed by gfgking.
</script>

Output

YES

Time Complexity: O(1)
Auxiliary Space: O(1)