style: remove interactive user input from extended_euclid_algorithm.cpp

[2501200112-Kunal]

Description

This Pull Request successfully refactors math/extended_euclid_algorithm.cpp to bring it into compliance with the repository's automated testing architecture. It removes legacy interactive terminal streams (std::cin), which block headless execution runners, and encapsulates the verification routines into clean, deterministic assert() validation states.

Changes Made

• Eliminated terminal input dependencies (std::cin) from the execution loop.
• Grouped edge cases and base executions into a dedicated static void test() suite.
• Validated correctness via automated standard assertions (<cassert>).
• Compiled and executed the update locally under the C++11 standard to ensure zero regression.

Closes # [Insert the Issue Number you created here]