Find the first and last M digits from K-th power of N

Given a two integers N and K, the task is to find the first M and last M digits of the number NK .
Examples: 

Input: N = 2345, K = 3, M = 3 
Output: 128 625 
Explanation: 
2345 3 = 12895213625 
Therefore, the first M(= 3) digits are 128 and the last three digits are 625.

Input: N = 12, K = 12, M = 4 
Output: 8916 8256 
Explanation: 
12 12 = 8916100448256 
 

Naive Approach: 
The simplest approach to solve the problem is to calculate the value of NK and iterate to first M digits and find the last M digits by NK mod 10M
Time Complexity: O(K) 
Auxiliary Space: O(1)

Efficient Approach: 
The above approach can be optimized by the following observation: 

Let us consider a number x which can be written as 10y Where y is a decimal number. 
Let x = NK
NK = 10 y 
Taking log10 on both sides of the above expression, we get: 
K * log10(N) = log10(10 y
=> K * log10(N) = y * (log1010) 
=> y = K * log10(N)
Now y will be a decimal number of form abc—.xyz— 
Therefore, 
NK = 10abc—.xyz— 
=> NK = 10abc— + 0.xyz— 
=> NK = 10abc— * 100.xyz— 
In the above equation, 10abc— only moves the decimal point forward. 
By calculating 100.xyz—, the first M digits can be figured out by moving the decimal point forward. 



Follow the steps below to solve the problem: 

  • Find the first M digits of NK by calculating (NK)mod(10M).
  • Calculate K * log10(N).
  • Calculate 10K * log10(N).
  • Find the first M digits by calculating 10K * log10(N) * 10M – 1 .
  • Print the first and last M digits obtained.

Below is the implementation of the above approach:

C++

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// C++ Program to implement
// the above approach
#include <bits/stdc++.h>
#define ll long long int
using namespace std;
  
// Function to find a^b modulo M
ll modPower(ll a, ll b, ll M)
{
    ll res = 1;
    while (b) {
        if (b & 1)
            res = res * a % M;
        a = a * a % M;
        b >>= 1;
    }
    return res;
}
  
// Function to find the first and last
// M digits from N^K
void findFirstAndLastM(ll N, ll K, ll M)
{
    // Calculate Last M digits
    ll lastM
        = modPower(N, K, (1LL) * pow(10, M));
  
    // Calculate First M digits
    ll firstM;
  
    double y = (double)K * log10(N * 1.0);
  
    // Extract the number after decimal
    y = y - (ll)y;
  
    // Find 10 ^ y
    double temp = pow(10.0, y);
  
    // Move the Decimal Point M - 1 digits forward
    firstM = temp * (1LL) * pow(10, M - 1);
  
    // Print the result
    cout << firstM << " " << lastM << endl;
}
  
// Driver Code
int main()
{
    ll N = 12, K = 12, M = 4;
  
    findFirstAndLastM(N, K, M);
    return 0;
}

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Java

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// Java program to implement
// the above approach
class GFG{
  
// Function to find a^b modulo M
static long modPower(long a, long b, long M)
{
    long res = 1;
    while (b > 0
    {
        if (b % 2 == 1)
            res = res * a % M;
              
        a = a * a % M;
        b >>= 1;
    }
    return res;
}
  
// Function to find the first and last
// M digits from N^K
static void findFirstAndLastM(long N, long K,
                              long M)
{
      
    // Calculate Last M digits
    long lastM = modPower(N, K, (1L) *
                 (long)Math.pow(10, M));
  
    // Calculate First M digits
    long firstM;
  
    double y = (double)K * Math.log10(N * 1.0);
  
    // Extract the number after decimal
    y = y - (long)y;
  
    // Find 10 ^ y
    double temp = Math.pow(10.0, y);
  
    // Move the Decimal Point M - 1 digits forward
    firstM = (long)(temp * (1L) *
             Math.pow(10, (M - 1)));
  
    // Print the result
    System.out.print(firstM + " " +
                      lastM + "\n");
}
  
// Driver Code
public static void main(String[] args)
{
    long N = 12, K = 12, M = 4;
  
    findFirstAndLastM(N, K, M);
}
}
  
// This code is contributed by Rajput-Ji

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C#

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// C# program to implement
// the above approach
using System;
using System.Collections.Generic;
  
class GFG{
  
// Function to find a^b modulo M
static long modPower(long a, long b, long M)
{
    long res = 1;
    while (b > 0) 
    {
        if (b % 2 == 1)
            res = res * a % M;
  
        a = a * a % M;
        b >>= 1;
    }
    return res;
}
  
// Function to find the first and last
// M digits from N^K
static void findFirstAndLastM(long N, long K,
                              long M)
{
  
    // Calculate Last M digits
    long lastM = modPower(N, K, (1L) * 
                 (long)Math.Pow(10, M));
  
    // Calculate First M digits
    long firstM;
  
    double y = (double)K * Math.Log10(N * 1.0);
  
    // Extract the number after decimal
    y = y - (long)y;
  
    // Find 10 ^ y
    double temp = Math.Pow(10.0, y);
  
    // Move the Decimal Point M - 1 digits forward
    firstM = (long)(temp * (1L) * 
             Math.Pow(10, (M - 1)));
  
    // Print the result
    Console.Write(firstM + " " +
                   lastM + "\n");
}
  
// Driver Code
public static void Main(String[] args)
{
    long N = 12, K = 12, M = 4;
  
    findFirstAndLastM(N, K, M);
}
}
  
// This code is contributed by gauravrajput1

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Output: 

8916 8256

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

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