Algorithms and Data Structures

# Binary Exponentiation

Given 3 numbers x, n, and m, you are asked to calculate the result of .

#### Input

The only line of the input contains 3 integers x, n, and m (1 ≤ x, n, m ≤ ).

#### Output

The program should print the result of the expression.

#### Examples

 Input Output 2 10 5 4

#### Explanation

1. 2^10 = 1024 ⇒ 1024 mod 5 = 4

### Tutorial

As all the numbers are pretty large, we need to use a fast algorithm to make sure it’s quick enough. 1. Binary exponentiation is an efficient way of computing large powers of a number. It reduces the number of multiplications required to compute the power, which makes it faster than the traditional method of looping from 1 through n and multiplying the result by x.

The binary exponentiation algorithm is based on the following mathematical observation: if we have a number x raised to the power of n, then can be represented as:
• if n is even
• if n is odd
This actually makes the process of calculating the way faster as we can calculate the once and then multiply the result twogether. So, the algorithm can look like this:
def binpow(x, n, m):
if n == 0:                        # Base case: x^0 = 1
return 1
if n % 2 == 0:                    # n is even => calculate half and multiply
half = binpow(x, n // 2, m)   # calculate the half
return (half * half) % m      # res = x^(n/2) * x^(n/2)

# n is odd => res = x * x^(n-1)
return (x * binpow(x, n - 1, m)) % m
With binary exponentiation, we reduce the time complexity from operations to as we split n on every iteration where n is even. It’s important to note that for each operation where n is odd, for the next iteration n is even.
Let’s simulate the algorithm on several inputs:
x = 2, n = 10 (we discard m for simplicity)
1. [n = 10] → n % 2 == 0 ⇒ calculate binpow(2, 5)
1. [n = 5] → n % 2 ≠ 0 ⇒ return 2 * binpow(2, 4)
1. [n = 4] → n % 2 == 0 ⇒ calculate binpow(2, 2)
1. [n = 2] → n % 2 == 0 ⇒ calculate binpow(2, 1)
1. [n = 1] ⇒ n % 2 ≠ 0 ⇒ return 2 * binpow(2, 0)
1. [n = 0] ⇒ n == 0 ⇒ return 1
1. [up to n = 1] ⇒ return 2 * 1 = 2
1. [up to n = 2] ⇒ return 2 * 2 = 4
1. [up to n = 4] ⇒ return 4 * 4 = 16
1. [up to n = 5] ⇒ return 2 * 16 ⇒ 32
1. [up to n = 10] ⇒ return 32 * 32 = 1024

#### Constraints

Time limit: 0.2 seconds

Memory limit: 512 MB

Output limit: 1 MB