How Can Euler\'s Theorem Help Calculate `pow(a, b) % MOD` for Large Fibonacci Numbers?

Euler's Theorem and Power Calculation

As you seek an efficient method to compute pow(a, b) % MOD in C where b can be a colossal Fibonacci number exceeding the capacity of the long long data type, we delve into Euler's theorem to provide an alternative solution.

Euler's totient function phi(MOD) plays a crucial role here. According to Euler's theorem, a^phi(MOD) equals 1 modulo MOD. This allows us to significantly reduce the calculation to a^(b % phi(MOD)). While finding phi(MOD) may require integer factorization techniques, it still eliminates the need for extensive power calculations.

Interestingly, Carmichael's function becomes pertinent in this scenario. By calculating lambda(MOD) (the Carmichael function), you can obtain the correct result for any a, b, and MOD.

Therefore, by utilizing Euler's theorem and its related functions, you can efficiently compute pow(a, b) % MOD even when b is a colossal value.

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