Find Last Digit of a^b for Large Numbers
You are given two integer numbers, the base a (number of digits d, such that 1 <= d <= 1000) and the index b (0 <= b <= 922*10^15). You have to find the last digit of a^b.
Examples:
Input : 3 10 Output : 9 Input : 6 2 Output : 6 Input : 150 53 Output : 0
After taking few examples, we can notice below pattern.
Number | Last digits that repeat in cycle 1 | 1 2 | 4, 8, 6, 2 3 | 9, 7, 1, 3 4 | 6, 4 5 | 5 6 | 6 7 | 9, 3, 1, 7 8 | 4, 2, 6, 8 9 | 1, 9
In the given table, we can see that maximum length for cycle repetition is 4.
Example: 2*2 = 4*2 = 8*2 = 16*2 = 32 last digit in 32 is 2 that means after multiplying 4 times digit repeat itself. So the algorithm is very simple .
Source : Brilliants.org
Algorithm :
- Since number are very large we store them as a string.
- Take last digit in base a.
- Now calculate b%4. Here b is very large.
- If b%4==0 that means b is completely divisible by 4, so our exponent now will be exp = 4 because by multiplying number 4 times, we get the last digit according to cycle table in above diagram.
- If b%4!=0 that means b is not completely divisible by 4, so our exponent now will be exp=b%4 because by multiplying number exponent times, we get the last digit according to cycle table in above diagram.
- Now calculate ldigit = pow( last_digit_in_base, exp ).
- Last digit of a^b will be ldigit%10.
Below is the implementation of above algorithm.
C++
// C++ code to find last digit of a^b #include <bits/stdc++.h> using namespace std; // Function to find b % a int Modulo( int a, char b[]) { // Initialize result int mod = 0; // calculating mod of b with a to make // b like 0 <= b < a for ( int i = 0; i < strlen (b); i++) mod = (mod * 10 + b[i] - '0' ) % a; return mod; // return modulo } // function to find last digit of a^b int LastDigit( char a[], char b[]) { int len_a = strlen (a), len_b = strlen (b); // if a and b both are 0 if (len_a == 1 && len_b == 1 && b[0] == '0' && a[0] == '0' ) return 1; // if exponent is 0 if (len_b == 1 && b[0] == '0' ) return 1; // if base is 0 if (len_a == 1 && a[0] == '0' ) return 0; // if exponent is divisible by 4 that means last // digit will be pow(a, 4) % 10. // if exponent is not divisible by 4 that means last // digit will be pow(a, b%4) % 10 int exp = (Modulo(4, b) == 0) ? 4 : Modulo(4, b); // Find last digit in 'a' and compute its exponent int res = pow (a[len_a - 1] - '0' , exp ); // Return last digit of result return res % 10; } // Driver program to run test case int main() { char a[] = "117" , b[] = "3" ; cout << LastDigit(a, b); return 0; } |
Java
// Java code to find last digit of a^b import java.io.*; import java.math.*; class GFG { // Function to find b % a static int Modulo( int a, char b[]) { // Initialize result int mod = 0 ; // calculating mod of b with a to make // b like 0 <= b < a for ( int i = 0 ; i < b.length; i++) mod = (mod * 10 + b[i] - '0' ) % a; return mod; // return modulo } // Function to find last digit of a^b static int LastDigit( char a[], char b[]) { int len_a = a.length, len_b = b.length; // if a and b both are 0 if (len_a == 1 && len_b == 1 && b[ 0 ] == '0' && a[ 0 ] == '0' ) return 1 ; // if exponent is 0 if (len_b == 1 && b[ 0 ] == '0' ) return 1 ; // if base is 0 if (len_a == 1 && a[ 0 ] == '0' ) return 0 ; // if exponent is divisible by 4 that means last // digit will be pow(a, 4) % 10. // if exponent is not divisible by 4 that means last // digit will be pow(a, b%4) % 10 int exp = (Modulo( 4 , b) == 0 ) ? 4 : Modulo( 4 , b); // Find last digit in 'a' and compute its exponent int res = ( int )(Math.pow(a[len_a - 1 ] - '0' , exp)); // Return last digit of result return res % 10 ; } // Driver program to run test case public static void main(String args[]) throws IOException { char a[] = "117" .toCharArray(), b[] = { '3' }; System.out.println(LastDigit(a, b)); } } // This code is contributed by Nikita Tiwari. |
# Python 3 code to find last digit of a ^ b
import math
# Function to find b % a
def Modulo(a, b) :
# Initialize result
mod = 0
# calculating mod of b with a to make
# b like 0 <= b < a
for i in range(0, len(b)) :
mod = (mod * 10 + (int)(b[i])) % a
return mod # return modulo
# function to find last digit of a ^ b
def LastDigit(a, b) :
len_a = len(a)
len_b = len(b)
# if a and b both are 0
if (len_a == 1 and len_b == 1 and b[0] == '0' and a[0] == '0') :
return 1
# if exponent is 0
if (len_b == 1 and b[0]=='0') :
return 1
# if base is 0
if (len_a == 1 and a[0] == '0') :
return 0
# if exponent is divisible by 4 that means last
# digit will be pow(a, 4) % 10.
# if exponent is not divisible by 4 that means last
# digit will be pow(a, b % 4) % 10
if((Modulo(4, b) == 0)) :
exp = 4
else :
exp = Modulo(4, b)
# Find last digit in 'a' and compute its exponent
res = math.pow((int)(a[len_a - 1]), exp)
# Return last digit of result
return res % 10
# Driver program to run test case
a = ['1', '1', '7']
b = ['3']
print(LastDigit(a, b))
# This code is contributed to Nikita Tiwari.
C#
// C# code to find last digit of a^b. using System; class GFG { // Function to find b % a static int Modulo( int a, char [] b) { // Initialize result int mod = 0; // calculating mod of b with a // to make b like 0 <= b < a for ( int i = 0; i < b.Length; i++) mod = (mod * 10 + b[i] - '0' ) % a; // return modulo return mod; } // Function to find last digit of a^b static int LastDigit( char [] a, char [] b) { int len_a = a.Length, len_b = b.Length; // if a and b both are 0 if (len_a == 1 && len_b == 1 && b[0] == '0' && a[0] == '0' ) return 1; // if exponent is 0 if (len_b == 1 && b[0] == '0' ) return 1; // if base is 0 if (len_a == 1 && a[0] == '0' ) return 0; // if exponent is divisible by 4 // that means last digit will be // pow(a, 4) % 10. if exponent is //not divisible by 4 that means last // digit will be pow(a, b%4) % 10 int exp = (Modulo(4, b) == 0) ? 4 : Modulo(4, b); // Find last digit in 'a' and // compute its exponent int res = ( int )(Math.Pow(a[len_a - 1] - '0' , exp)); // Return last digit of result return res % 10; } // Driver program to run test case public static void Main() { char [] a = "117" .ToCharArray(), b = { '3' }; Console.Write(LastDigit(a, b)); } } // This code is contributed by nitin mittal. |
PHP
<?php // php code to find last digit of a^b // Function to find b % a function Modulo( $a , $b ) { // Initialize result $mod = 0; // calculating mod of b with a to make // b like 0 <= b < a for ( $i = 0; $i < strlen ( $b ); $i ++) $mod = ( $mod * 10 + $b [ $i ] - '0' ) % $a ; return $mod ; // return modulo } // function to find last digit of a^b function LastDigit( $a , $b ) { $len_a = strlen ( $a ); $len_b = strlen ( $b ); // if a and b both are 0 if ( $len_a == 1 && $len_b == 1 && $b [0] == '0' && $a [0] == '0' ) return 1; // if exponent is 0 if ( $len_b == 1 && $b [0] == '0' ) return 1; // if base is 0 if ( $len_a == 1 && $a [0] == '0' ) return 0; // if exponent is divisible by 4 that // means last digit will be pow(a, 4) // % 10. if exponent is not divisible // by 4 that means last digit will be // pow(a, b%4) % 10 $exp = (Modulo(4, $b ) == 0) ? 4 : Modulo(4, $b ); // Find last digit in 'a' and compute // its exponent $res = pow( $a [ $len_a - 1] - '0' , $exp ); // Return last digit of result return $res % 10; } // Driver program to run test case $a = "117" ; $b = "3" ; echo LastDigit( $a , $b ); // This code is contributed by nitin mittal. ?> |
Javascript
<script> // Javascript code to find last digit of a^b // Function to find b % a function Modulo(a, b) { // Initialize result let mod = 0; // calculating mod of b with a to make // b like 0 <= b < a for (let i = 0; i < b.length; i++) mod = (mod * 10 + b[i] - '0' ) % a; return mod; // return modulo } // function to find last digit of a^b function LastDigit(a, b) { let len_a = a.length; let len_b = b.length; // if a and b both are 0 if (len_a == 1 && len_b == 1 && b[0] == '0' && a[0] == '0' ) return 1; // if exponent is 0 if (len_b == 1 && b[0] == '0' ) return 1; // if base is 0 if (len_a == 1 && a[0] == '0' ) return 0; // if exponent is divisible by 4 that // means last digit will be pow(a, 4) // % 10. if exponent is not divisible // by 4 that means last digit will be // pow(a, b%4) % 10 exp = (Modulo(4, b) == 0) ? 4 : Modulo(4, b); // Find last digit in 'a' and compute // its exponent res = Math.pow(a[len_a - 1] - '0' , exp); // Return last digit of result return res % 10; } // Driver program to run test case let a = "117" ; let b = "3" ; document.write(LastDigit(a, b)); // This code is contributed by _saurabh_jaiswal </script> |
Output :
3
This article is contributed by Shashank Mishra ( Gullu ). This article is reviewed by team geeksforgeeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.