Dados dos números sa y sb representados como strings, encuentre a b % MOD donde MOD es 1e9 + 7. Los números a y b pueden contener hasta 10 6 dígitos.
Ejemplos:
Entrada: sa = 2, sb = 3
Salida: 8
Entrada: sa = 10000000000000000000000000000000000000000000
sb = 10000000000000000000000000000000000000000000
Salida: 4 546233
Como a y b muy grandes (pueden contener hasta 10^6 dígitos cada uno). Entonces, lo que podemos hacer, aplicamos el pequeño teorema de Fermat y la propiedad del módulo para reducir a y b .
Reducir a:
Como sabemos,
(ab) % MOD = ((a % MOD)b) % MOD
Reducir b:
cómo reducir b , ya lo hemos discutido en Find (a^b)%m donde ‘b’ es muy grande
. Ahora finalmente tenemos que tanto a como b están en el rango de 1<=a, b<=10^ 9+7. Por lo tanto, ahora podemos usar nuestra exponenciación modular para calcular la respuesta requerida.
C++
// CPP program to find (a^b) % MOD where a and
// b may be very large and represented as strings.
#include <bits/stdc++.h>
using namespace std;
#define ll long long int
const ll MOD = 1e9 + 7;
// Returns modulo exponentiation for two numbers
// represented as long long int. It is used by
// powerStrings(). Its complexity is log(n)
ll powerLL(ll x, ll n)
{
ll result = 1;
while (n) {
if (n & 1)
result = result * x % MOD;
n = n / 2;
x = x * x % MOD;
}
return result;
}
// Returns modulo exponentiation for two numbers
// represented as strings. It is used by
// powerStrings()
ll powerStrings(string sa, string sb)
{
// We convert strings to number
ll a = 0, b = 0;
// calculating a % MOD
for (int i = 0; i < sa.length(); i++)
a = (a * 10 + (sa[i] - '0')) % MOD;
// calculating b % (MOD - 1)
for (int i = 0; i < sb.length(); i++)
b = (b * 10 + (sb[i] - '0')) % (MOD - 1);
// Now a and b are long long int. We
// calculate a^b using modulo exponentiation
return powerLL(a, b);
}
int main()
{
// As numbers are very large
// that is it may contains upto
// 10^6 digits. So, we use string.
string sa = "2", sb = "3";
cout << powerStrings(sa, sb) << endl;
return 0;
}
Java
// Java program to find (a^b) % MOD
// where a and b may be very large
// and represented as strings.
import java.util.*;
class GFG
{
static long MOD = (long) (1e9 + 7);
// Returns modulo exponentiation for two numbers
// represented as long long int. It is used by
// powerStrings(). Its complexity is log(n)
static long powerLL(long x, long n)
{
long result = 1;
while (n > 0)
{
if (n % 2 == 1)
{
result = result * x % MOD;
}
n = n / 2;
x = x * x % MOD;
}
return result;
}
// Returns modulo exponentiation for
// two numbers represented as strings.
// It is used by powerStrings()
static long powerStrings(String sa, String sb)
{
// We convert strings to number
long a = 0, b = 0;
// calculating a % MOD
for (int i = 0; i < sa.length(); i++)
{
a = (a * 10 + (sa.charAt(i) - '0')) %
MOD;
}
// calculating b % (MOD - 1)
for (int i = 0; i < sb.length(); i++)
{
b = (b * 10 + (sb.charAt(i) - '0')) %
(MOD - 1);
}
// Now a and b are long long int. We
// calculate a^b using modulo exponentiation
return powerLL(a, b);
}
// Driver code
public static void main(String[] args)
{
// As numbers are very large
// that is it may contains upto
// 10^6 digits. So, we use string.
String sa = "2", sb = "3";
System.out.println(powerStrings(sa, sb));
}
}
// This code is contributed by Rajput-JI
Python3
# Python3 program to find (a^b) % MOD
# where a and b may be very large
# and represented as strings.
MOD = 1000000007;
# Returns modulo exponentiation
# for two numbers represented as
# long long int. It is used by
# powerStrings(). Its complexity
# is log(n)
def powerLL(x, n):
result = 1;
while (n):
if (n & 1):
result = result * x % MOD;
n = int(n / 2);
x = x * x % MOD;
return result;
# Returns modulo exponentiation
# for two numbers represented as
# strings. It is used by powerStrings()
def powerStrings(sa, sb):
# We convert strings to number
a = 0;
b = 0;
# calculating a % MOD
for i in range(len(sa)):
a = (a * 10 + (ord(sa[i]) -
ord('0'))) % MOD;
# calculating b % (MOD - 1)
for i in range(len(sb)):
b = (b * 10 + (ord(sb[i]) -
ord('0'))) % (MOD - 1);
# Now a and b are long long int.
# We calculate a^b using modulo
# exponentiation
return powerLL(a, b);
# Driver code
# As numbers are very large
# that is it may contains upto
# 10^6 digits. So, we use string.
sa = "2";
sb = "3";
print(powerStrings(sa, sb));
# This code is contributed by mits
C#
// C# program to find (a^b) % MOD where a and b
// may be very large and represented as strings.
using System;
class GFG
{
static long MOD = (long) (1e9 + 7);
// Returns modulo exponentiation for two numbers
// represented as long long int. It is used by
// powerStrings(). Its complexity is log(n)
static long powerLL(long x, long n)
{
long result = 1;
while (n > 0)
{
if (n % 2 == 1)
{
result = result * x % MOD;
}
n = n / 2;
x = x * x % MOD;
}
return result;
}
// Returns modulo exponentiation for
// two numbers represented as strings.
// It is used by powerStrings()
static long powerStrings(String sa, String sb)
{
// We convert strings to number
long a = 0, b = 0;
// calculating a % MOD
for (int i = 0; i < sa.Length; i++)
{
a = (a * 10 + (sa[i] - '0')) % MOD;
}
// calculating b % (MOD - 1)
for (int i = 0; i < sb.Length; i++)
{
b = (b * 10 + (sb[i] - '0')) % (MOD - 1);
}
// Now a and b are long long int. We
// calculate a^b using modulo exponentiation
return powerLL(a, b);
}
// Driver code
public static void Main(String[] args)
{
// As numbers are very large
// that is it may contains upto
// 10^6 digits. So, we use string.
String sa = "2", sb = "3";
Console.WriteLine(powerStrings(sa, sb));
}
}
// This code is contributed by 29AjayKumar
PHP
<?php
// PHP program to find (a^b) % MOD
// where a and b may be very large
// and represented as strings.
$MOD = 1000000007;
// Returns modulo exponentiation
// for two numbers represented as
// long long int. It is used by
// powerStrings(). Its complexity
// is log(n)
function powerLL($x, $n)
{
global $MOD;
$result = 1;
while ($n)
{
if ($n & 1)
$result = $result * $x % $MOD;
$n = (int)$n / 2;
$x = $x * $x % $MOD;
}
return $result;
}
// Returns modulo exponentiation
// for two numbers represented as
// strings. It is used by powerStrings()
function powerStrings($sa, $sb)
{
global $MOD;
// We convert strings to number
$a = 0;
$b = 0;
// calculating a % MOD
for ($i = 0; $i < strlen($sa); $i++)
$a = ($a * 10 + ($sa[$i] -
'0')) % $MOD;
// calculating b % (MOD - 1)
for ($i = 0; $i < strlen($sb); $i++)
$b = ($b * 10 + ($sb[$i] - '0')) %
($MOD - 1);
// Now a and b are long long int.
// We calculate a^b using modulo
// exponentiation
return powerLL($a, $b);
}
// Driver code
// As numbers are very large
// that is it may contains upto
// 10^6 digits. So, we use string.
$sa = "2";
$sb = "3";
echo powerStrings($sa, $sb);
// This code is contributed by mits
?>
Javascript
<script>
// Javascript program to find (a^b) % MOD where a and b
// may be very large and represented as strings.
let MOD = (1e9 + 7);
// Returns modulo exponentiation for two numbers
// represented as long long let. It is used by
// powerStrings(). Its complexity is log(n)
function powerLL(x, n)
{
let result = 1;
while (n > 0)
{
if (n % 2 == 1)
{
result = result * x % MOD;
}
n = Math.floor(n / 2);
x = x * x % MOD;
}
return result;
}
// Returns modulo exponentiation for
// two numbers represented as strings.
// It is used by powerStrings()
function powerStrings(sa, sb)
{
// We convert strings to number
let a = 0, b = 0;
// calculating a % MOD
for (let i = 0; i < sa.length; i++)
{
a = (a * 10 + (sa[i] - '0')) % MOD;
}
// calculating b % (MOD - 1)
for (let i = 0; i < sb.length; i++)
{
b = (b * 10 + (sb[i] - '0')) % (MOD - 1);
}
// Now a and b are long long let. We
// calculate a^b using modulo exponentiation
return powerLL(a, b);
}
// Driver Code
// As numbers are very large
// that is it may contains upto
// 10^6 digits. So, we use string.
let sa = "2", sb = "3";
document.write(powerStrings(sa, sb));
</script>
Producción:
8
Publicación traducida automáticamente
Artículo escrito por sahilshelangia y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA