GCD of a number raised to some power and another number

Given three numbers a, b, n. Find GCD(an, b).

Examples:

Input : a = 2, b = 3, n = 3
Output : 1
2^3 = 8. GCD of 8 and 3 is 1. 

Input : a = 2, b = 4, n = 5
Output : 4

First Approach : Brute Force approach is to first compute a^n, then compute GCD of a^n and b.

C++

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// CPP program to find GCD of a^n and b.
#include <bits/stdc++.h>
using namespace std;
  
typedef long long int ll;
  
ll gcd(ll a, ll b)
{
    if (a == 0)
        return b;
    return gcd(b % a, a);
}
  
// Returns GCD of a^n and b.
ll powGCD(ll a, ll n, ll b)
{
    for (int i = 0; i < n; i++)
        a = a * a;
  
    return gcd(a, b);
}
  
// Driver code
int main()
{
    ll a = 10, b = 5, n = 2;
    cout << powGCD(a, n, b);
    return 0;
}

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Java

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// Java program to find GCD of a^n and b.
  
import java.io.*;
  
class GFG {
  
  
static long gcd(long a, long b)
{
    if (a == 0)
        return b;
    return gcd(b % a, a);
}
  
// Returns GCD of a^n and b.
static long powGCD(long a, long n, long b)
{
    for (int i = 0; i < n; i++)
        a = a * a;
  
    return gcd(a, b);
}
  
// Driver code
    public static void main (String[] args) {
    long a = 10, b = 5, n = 2;
    System.out.println(powGCD(a, n, b));
    }
}
// This code is contributed by anuj_67..

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Python3

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# Python 3 program to find 
# GCD of a^n and b.
def gcd(a, b):
    if (a == 0):
        return b
    return gcd(b % a, a)
  
# Returns GCD of a^n and b.
def powGCD(a, n, b):
    for i in range(0, n + 1, 1):
        a = a * a
  
    return gcd(a, b)
  
# Driver code
if __name__ == '__main__':
    a = 10
    b = 5
    n = 2
    print(powGCD(a, n, b))
      
# This code is contributed 
# by Surendra_Gangwar

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C#

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// C# program to find GCD of a^n and b. 
using System;
  
class GFG
{
public static long gcd(long a, long b)
{
    if (a == 0)
    {
        return b;
    }
    return gcd(b % a, a);
}
  
// Returns GCD of a^n and b. 
public static long powGCD(long a, 
                          long n, long b)
{
    for (int i = 0; i < n; i++)
    {
        a = a * a;
    }
  
    return gcd(a, b);
}
  
// Driver code 
public static void Main(string[] args)
{
    long a = 10, b = 5, n = 2;
    Console.WriteLine(powGCD(a, n, b));
}
}
  
// This code is contributed 
// by Shrikant13

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PHP

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<?php
// PHP program to find GCD of a^n and b
function gcd($a, $b)
{
    if ($a == 0)
        return $b;
    return gcd($b % $a, $a);
}
  
// Returns GCD of a^n and b.
function powGCD($a, $n, $b)
{
    for ($i = 0; $i < $n; $i++)
        $a = $a * $a;
  
    return gcd($a, $b);
}
  
// Driver code
$a = 10;
$b = 5;
$n = 2;
  
echo powGCD($a, $n, $b);
  
// This code is contributed by ANKITRAI1
?>

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Output:

5

But, what if n is very large (say > 10^9). Modular Exponentiation is the way. We know (a*b) % m = ( (a%m) * (b%m) ) % m). We also know gcd(a, b) = gcd(b%a, a) . So instead of computing ” pow(a, n), we use modular exponentiation.

C++

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// C++ program of the above approach
#include <bits/stdc++.h>
using namespace std;
  
typedef long long int ll;
  
/* Calculates modular exponentiation, i.e.,
   (x^y)%p in O(log y) */
ll power(ll x, ll y, ll p)
{
    ll 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;
}
  
  
ll gcd(ll a, ll b)
{
    if (a == 0)
        return b;
    return gcd(b % a, a);
}
  
// Returns GCD of a^n and b
ll powerGCD(ll a, ll b, ll n)
{
    ll e = power(a, n, b);
    return gcd(e, b);
}
  
// Driver code
int main()
{
    ll a = 5, b = 4, n = 2;
    cout << powerGCD(a, b, n);
    return 0;
}

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Java

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// Java program of the above approach 
import java.util.*;
class Solution{
    
    
/* Calculates modular exponentiation, i.e., 
   (x^y)%p in O(log y) */
static long 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 res; 
    
    
static long gcd(long a, long b) 
    if (a == 0
        return b; 
    return gcd(b % a, a); 
    
// Returns GCD of a^n and b 
static long powerGCD(long a, long b, long n) 
    long e = power(a, n, b); 
    return gcd(e, b); 
    
// Driver code 
public static void main(String args[])
    long a = 5, b = 4, n = 2
    System.out.print( powerGCD(a, b, n)); 
  
}
//contributed by Arnab Kundu

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Python3

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# Python3 program of the above approach
   
# Calculates modular exponentiation, i.e.,
 # (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
   
   
def gcd(a,  b):
  
    if (a == 0):
        return b
    return gcd(b % a, a)
   
# Returns GCD of a^n and b
def powerGCD( a,  b,  n):
  
    e = power(a, n, b)
    return gcd(e, b)
   
# Driver code
if __name__ == "__main__":
  
    a = 5
    b = 4
    n = 2
    print (powerGCD(a, b, n))

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C#

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// C# program of the above approach 
using System;
class GFG
{
  
/* Calculates modular exponentiation, 
i.e.,  (x^y)%p in O(log y) */
static long 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 res; 
  
static long gcd(long a, long b) 
    if (a == 0) 
        return b; 
    return gcd(b % a, a); 
  
// Returns GCD of a^n and b 
static long powerGCD(long a, long b,
                             long n) 
    long e = power(a, n, b); 
    return gcd(e, b); 
  
// Driver code 
public static void Main()
    long a = 5, b = 4, n = 2; 
    Console.Write( powerGCD(a, b, n)); 
  
// This code is contributed 
// by Akanksha Rai

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PHP

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<?php
// PHP program of the above approach 
// Calculates modular exponentiation, 
// i.e.,(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 gcd ($a, $b
    if ($a == 0) 
        return $b
    return gcd($b % $a, $a); 
  
// Returns GCD of a^n and b 
function powerGCD($a, $b, $n
    $e = power($a, $n, $b); 
    return gcd($e, $b); 
  
// Driver code 
$a = 5;
$b = 4;
$n = 2; 
echo powerGCD($a, $b, $n); 
  
// This code is contributed by Sachin.
?>

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Output:

1


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