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Print last k digits of a^b (a raised to power b)

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  • Difficulty Level : Medium
  • Last Updated : 22 Nov, 2021

Given positive integers k, a and b we need to print last k digits of a^b ie.. pow(a, b).

Input Constraint:
k <= 9, a <= 10^6, b<= 10^6

Examples: 

Input : a = 11, b = 3, k = 2
Output : 31
Explanation : a^b = 11^3 = 1331, hence 
last two digits are 31

Input : a = 10, b = 10000, k = 5
Output : 00000
Explanation : a^b = 1000..........0 (total 
zeros = 10000), hence last 5 digits are 00000

Naive Solution

First Calculate a^b, then take last k digits by taking modulo with 10^k. Above solution fails when a^b is too large, as we can hold at most 2^64 -1 in C/C++.

Efficient Solution

The efficient way is to keep only k digits after every multiplication. This idea is very similar to discussed in Modular Exponentiation where we discussed a general way to find (a^b)%c, here in this case c is 10^k.

Here is the implementation.

C++




// C++ code to find last k digits of a^b
#include <iostream>
using namespace std;
 
/* Iterative Function to calculate (x^y)%p in O(log y) */
int power(long long int x, long long int y, long long int p)
{
    long long int res = 1; // Initialize result
 
    x = x % p; // Update x if it is more than or
    // equal to p
 
    while (y > 0) {
 
        // If y is odd, multiply x with result
        if (y & 1)
            res = (res * x) % p;
 
        // y must be even now
        y = y >> 1; // y = y/2
        x = (x * x) % p;
    }
    return res;
}
 
// C++ function to calculate
// number of digits in x
int numberOfDigits(int x)
{
    int i = 0;
    while (x) {
        x /= 10;
        i++;
    }
    return i;
}
 
// C++ function to print last k digits of a^b
void printLastKDigits(int a, int b, int k)
{
    cout << "Last " << k;
    cout << " digits of " << a;
    cout << "^" << b << " = ";
     
    // Generating 10^k
    int temp = 1;
    for (int i = 1; i <= k; i++)
        temp *= 10;
 
    // Calling modular exponentiation
    temp = power(a, b, temp);
 
    // Printing leftmost zeros. Since (a^b)%k
    // can have digits less then k. In that
    // case we need to print zeros
    for (int i = 0; i < k - numberOfDigits(temp); i++)
        cout << 0;
 
    // If temp is not zero then print temp
    // If temp is zero then already printed
    if (temp)
        cout << temp;
}
 
// Driver program to test above functions
int main()
{
    int a = 11;
    int b = 3;
    int k = 2;
    printLastKDigits(a, b, k);
    return 0;
}

Java




// Java code to find last k digits of a^b
 
public class GFG
{
    /* Iterative Function to calculate (x^y)%p in O(log y) */
    static int power(long x, long y, long p)
    {
        long res = 1; // Initialize result
      
        x = x % p; // Update x if it is more than or
        // equal to p
      
        while (y > 0) {
      
            // If y is odd, multiply x with result
            if ((y & 1) != 0)
                res = (res * x) % p;
      
            // y must be even now
            y = y >> 1; // y = y/2
            x = (x * x) % p;
        }
        return (int) res;
    }
      
    // Method  to print last k digits of a^b
    static void printLastKDigits(int a, int b, int k)
    {
        System.out.print("Last " + k + " digits of " + a +
                            "^"  +  b + " = ");
          
        // Generating 10^k
        int temp = 1;
        for (int i = 1; i <= k; i++)
            temp *= 10;
      
        // Calling modular exponentiation
        temp = power(a, b, temp);
      
        // Printing leftmost zeros. Since (a^b)%k
        // can have digits less then k. In that
        // case we need to print zeros
        for (int i = 0; i < k - Integer.toString(temp).length() ; i++)
            System.out.print(0);
      
        // If temp is not zero then print temp
        // If temp is zero then already printed
        if (temp != 0)
            System.out.print(temp);
    }
     
    // Driver Method
    public static void main(String[] args)
    {
        int a = 11;
        int b = 3;
        int k = 2;
        printLastKDigits(a, b, k);
    }
}

Python3




# Python 3 code to find last
# k digits of a^b
 
# Iterative Function to calculate
# (x^y)%p in O(log y)
def power(x, y, p):
 
    res = 1 # Initialize result
 
    x = x % p # Update x if it is more
              # than or equal to p
 
    while (y > 0) :
 
        # If y is odd, multiply
        # x with result
        if (y & 1):
            res = (res * x) % p
 
        # y must be even now
        y = y >> 1 # y = y/2
        x = (x * x) % p
     
    return res
 
# function to calculate
# number of digits in x
def numberOfDigits(x):
 
    i = 0
    while (x) :
        x //= 10
        i += 1
 
    return i
 
# function to print last k digits of a^b
def printLastKDigits( a, b, k):
 
    print("Last " + str( k)+" digits of " +
                    str(a) + "^" + str(b), end = " = ")
     
    # Generating 10^k
    temp = 1
    for i in range(1, k + 1):
        temp *= 10
 
    # Calling modular exponentiation
    temp = power(a, b, temp)
 
    # Printing leftmost zeros. Since (a^b)%k
    # can have digits less then k. In that
    # case we need to print zeros
    for i in range( k - numberOfDigits(temp)):
        print("0")
 
    # If temp is not zero then print temp
    # If temp is zero then already printed
    if (temp):
        print(temp)
 
# Driver Code
if __name__ == "__main__":
    a = 11
    b = 3
    k = 2
    printLastKDigits(a, b, k)
 
# This code is contributed
# by ChitraNayal

C#




// C# code to find last k digits of a^b
using System;
 
class GFG
{
 
// Iterative Function to calculate
// (x^y)%p in O(log y)
static int power(long x, long y, long p)
{
    long res = 1; // Initialize result
 
    x = x % p; // Update x if it is more
               // than or equal to p
 
    while (y > 0)
    {
 
        // If y is odd, multiply x with result
        if ((y & 1) != 0)
            res = (res * x) % p;
 
        // y must be even now
        y = y >> 1; // y = y/2
        x = (x * x) % p;
    }
    return (int) res;
}
 
// Method to print last k digits of a^b
static void printLastKDigits(int a, int b, int k)
{
    Console.Write("Last " + k + " digits of " +
                            a + "^" + b + " = ");
     
    // Generating 10^k
    int temp = 1;
    for (int i = 1; i <= k; i++)
        temp *= 10;
 
    // Calling modular exponentiation
    temp = power(a, b, temp);
 
    // Printing leftmost zeros. Since (a^b)%k
    // can have digits less then k. In that
    // case we need to print zeros
    for (int i = 0;    
             i < k - temp.ToString().Length; i++)
        Console.WriteLine(0);
 
    // If temp is not zero then print temp
    // If temp is zero then already printed
    if (temp != 0)
        Console.Write(temp);
}
 
// Driver Code
public static void Main()
{
    int a = 11;
    int b = 3;
    int k = 2;
    printLastKDigits(a, b, k);
}
}
 
// This code is contributed
// by 29AjayKumar

PHP




<?php
// PHP code to find last k digits of a^b
 
/* Iterative Function to calculate
   (x^y)%p in O(log y) */
 
function power($x, $y, $p)
{
    $res = 1; // Initialize result
 
    $x = $x % $p; // Update x if it is more
                  // than or equal to p
 
    while ($y > 0)
    {
 
        // If y is odd, multiply x
        // with result
        if ($y & 1)
            $res = ($res * $x) % $p;
 
        // y must be even now
        $y = $y >> 1; // y = y/2
        $x = ($x * $x) % $p;
    }
    return $res;
}
 
// function to calculate
// number of digits in x
function numberOfDigits($x)
{
    $i = 0;
    while ($x)
    {
        $x = (int)$x / 10;
        $i++;
    }
    return $i;
}
 
// function to print last k digits of a^b
function printLastKDigits( $a, $b, $k)
{
    echo "Last ",$k;
    echo " digits of " ,$a;
    echo "^" , $b , " = ";
     
    // Generating 10^k
    $temp = 1;
    for ($i = 1; $i <= $k; $i++)
        $temp *= 10;
 
    // Calling modular exponentiation
    $temp = power($a, $b, $temp);
 
    // Printing leftmost zeros. Since
    // (a^b)%k can have digits less
    // then k. In that case we need
    // to print zeros
    for ($i = 0;
         $i < $k - numberOfDigits($temp); $i++)
        echo 0;
 
    // If temp is not zero then print temp
    // If temp is zero then already printed
    if ($temp)
        echo $temp;
}
 
// Driver Code
$a = 11;
$b = 3;
$k = 2;
printLastKDigits($a, $b, $k);
 
// This code is contributed by ajit
?>

Javascript




<script>
// javascript code to find last k digits of a^b
 
    /* Iterative Function to calculate (x^y)%p in O(log y) */
    function power(x , y , p) {
        var res = 1; // Initialize result
 
        x = x % p; // Update x if it is more than or
        // equal to p
 
        while (y > 0) {
 
            // If y is odd, multiply x with result
            if ((y & 1) != 0)
                res = (res * x) % p;
 
            // y must be even now
            y = y >> 1; // y = y/2
            x = (x * x) % p;
        }
        return parseInt( res);
    }
 
    // Method to print last k digits of a^b
    function printLastKDigits(a , b , k) {
        document.write("Last " + k + " digits of " + a + "^" + b + " = ");
 
        // Generating 10^k
        var temp = 1;
        for (i = 1; i <= k; i++)
            temp *= 10;
 
        // Calling modular exponentiation
        temp = power(a, b, temp);
 
        // Printing leftmost zeros. Since (a^b)%k
        // can have digits less then k. In that
        // case we need to print zeros
        for (i = 0; i < k - temp.toString().length; i++)
            document.write(0);
 
        // If temp is not zero then print temp
        // If temp is zero then already printed
        if (temp != 0)
            document.write(temp);
    }
 
    // Driver Method
     
        var a = 11;
        var b = 3;
        var k = 2;
        printLastKDigits(a, b, k);
 
// This code contributed by gauravrajput1
</script>

Output: 
 

Last 2 digits of 11^3 = 31

Time Complexity : O(log b) 
Space Complexity : O(1) 
This article is contributed by Pratik Chhajer. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 


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