How Can Euler\'s Theorem and the Totient Function Efficiently Calculate pow(a, b) % MOD with Large \'b\'?

Calculating the Power of a Number with Exponentiation Constraints

In calculating pow(a, b) % MOD, where 'b' can be extremely large and not representable in traditional data types, a more efficient approach is required to handle such exponential constraints.

The Euler's theorem and totient function provide a key insight into solving this problem. The Euler's theorem states that pow(a, b) % MOD is equivalent to pow(a, b % phi(MOD)) % MOD, where 'phi(MOD)' is the Euler's totient function that counts the number of positive integers less than 'MOD' that are relatively prime to it.

To determine 'phi(MOD)', several methods can be employed, including integer factorization and the Carmichael function. Understanding the relationship between the power of 'a' and the remainder after division by 'phi(MOD)' allows for efficient calculation of the desired value.

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