计算非常大的指数的最佳方法是什么？

Question

I'm new to Python, and I'm trying to solve a programming problem where I have to compute 5 to the n ^th power, and once I have that, I just have to output the last two digits of that number. This is the code I wrote below:

print(str(pow(5, int(input())))[-2:])

The code works fine, for the most part, but exceeds the 500 ms time limit when the input is a large number like 1000000000000000000

What is the most efficient way to process such large inputs like this as an exponent without exceeding the time limit?

Answer 1

It can be shown by induction that 5^n mod 100 = 25, for all n >= 2. This is clear when n = 2. Suppose 5^n is of the form 100k+25. Then 5^(n+1) = 100(5k+1)+25, whence 5^(n+1) mod 100 = 25. Hence, the last two digits of 5^n is 25, for all n >= 2.

Some general tricks for computing a^n mod b efficiently include repeated squaring for computing a^n, and computing remainders in each step so that the numbers stay small.

Answer 2

For these kinds of heavy calculations you can use Multi Processing to use CPU cores to break down the calculation to little calculation simultaneously. For example (dummy example) we know 5^4 equals to 625. So we can use 2 cores of CPU to calculate 5^2, then multiply the result. 5^2 * 5^2 = 625.