Representation of a number in powers of other

Given two numbers w and m, we need to determine whether it is possible to represent m in terms of powers of w. The powers of number w can be added or subtracted to obtain m and each powers of w can be used only once .

Examples:

Input : 3 7
Output : Yes
As 7 = 9 - 3 + 1 (3^2 - 3^1 + 3^0 )
so it is possible .

Input : 100 50
Output : No
As 50 is less than 100 so we can never
represent it in the powers of 100 .

Here we have to represent m in terms of powers of w used only once so it can be shown through the following equation .
c0 + c1*w^1 + c2*w^2 + … = m —— (Equation 1)

Where each c0, c1, c2 … are either -1 (for subtracting that power of w ), 0 (not using that power of w ), 1 (for adding that power of w ) .

=> c1*w^1 + c2*w^2 + … = m – c0
=> w(c1 + c2*w^1 + c3*w^2 + … ) = m – c0
=> c1 + c2*w^1 + c3*w^2 + … = (m – c0)/w —— (Equation 2)

Now, notice equation 1 and equation 2 — we are trying to solve the same problem all over again. So we have to recurse till m > 0 . For such a solution to exist (m — ci) must be a multiple of w, where ci is the coefficient of the equation . The ci can be -1, 0, 1 . So we have to check for all three possibilities ( ( m – 1 ) % w == 0), ( ( m + 1 ) % w == 0) and ( m % w == 0) . If it is not, then there will not be any solution.

C++

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// CPP program to check if m can be represented
// as powers of w.
#include <bits/stdc++.h>
using namespace std;
  
bool asPowerSum(int w, int m)
{
    while (m) {
        if ((m - 1) % w == 0) 
            m = (m - 1) / w;
       else if ((m + 1) % w == 0) 
            m = (m + 1) / w;
          
        else if (m % w == 0) 
            m = m / w;
          
        else
            break; // None of 3 worked.
    }
  
    // If m is not zero means, it can't be 
    // represented in terms of powers of w.
    return (m == 0);
}
  
// Driver code
int main()
{
    int w = 3, m = 7;
    if (asPowerSum(w, m))
        cout << "Yes" << endl;
    else
        cout << "No" << endl;
   return 0;
}

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Java

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// Java program to check if m can 
// be represented as powers of w.
  
class GFG
{
    static boolean asPowerSum(int w, int m)
    {
        while (m > 0
        {
            if ((m - 1) % w == 0
                m = (m - 1) / w;
              
            else if ((m + 1) % w == 0
                m = (m + 1) / w;
              
            else if (m % w == 0
                m = m / w;
              
            else
                break; // None of 3 worked.
        }
      
        // If m is not zero means, it can't be 
        // represented in terms of powers of w.
        return (m == 0);
    
      
    // Driver function
    public static void main (String[] args)
    {
        int w = 3, m = 7;
        if (asPowerSum(w, m))
            System.out.println("Yes");
        else
            System.out.println("No"); 
    }
}
  
// This code is contributed by Anant Agarwal.

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Python3

# Python3 program to check if m can
# be represented as powers of w.
def asPowerSum(w, m):
while (m > 0):
if ((m – 1) % w == 0):
m = (m – 1) / w;

elif ((m + 1) % w == 0):
m = (m + 1) / w;

elif (m % w == 0):
m = m / w;

else:
break; # None of 3 worked.

# If m is not zero means, it can’t be
# represented in terms of powers of w.
return (m == 0);

# Driver code
w = 3;
m = 7;
if (asPowerSum(w, m)):
print(“Yes”);
else:
print(“No”);

# This code is contributed by mits

C#

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// C# program to check if 
// m can be represented
// as powers of w.
using System;
  
class GFG
{
    static bool asPowerSum(int w, 
                           int m)
    {
        while (m > 0) 
        {
            if ((m - 1) % w == 0) 
                m = (m - 1) / w;
              
            else if ((m + 1) % w == 0) 
                m = (m + 1) / w;
              
            else if (m % w == 0) 
                m = m / w;
              
            else
                break; // None of 3 worked.
        }
      
        // If m is not zero means, 
        // it can't be represented
        // in terms of powers of w.
        return (m == 0);
    
      
    // Driver Code
    static public void Main ()
    {
        int w = 3, m = 7;
        if (asPowerSum(w, m))
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No"); 
    }
}
  
// This code is contributed 
// by akt_mit.

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PHP

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<?php
// PHP program to check if m can 
// be represented as powers of w.
  
function asPowerSum($w, $m)
{
    while ($m
    {
        if (($m - 1) % $w == 0) 
            $m = ($m - 1) / $w;
    else if (($m + 1) % $w == 0) 
            $m = ($m + 1) / $w;
          
        else if ($m % $w == 0) 
            $m = $m / $w;
          
        else
            break; // None of 3 worked.
    }
  
    // If m is not zero means, it can't be 
    // represented in terms of powers of w.
    return ($m == 0);
}
  
// Driver code
$w = 3; 
$m = 7;
if (asPowerSum($w, $m))
    echo "Yes\n";
else
    echo "No\n";
  
// This code is contributed by mits
?>

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Output:

Yes


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