Quantum Core Frequency Modulation

🎯 Level: Medium ⭐ Score: 50 ⏱️ Estimated Time: 30 mins
📜 Problem Description

In the cutting-edge field of quantum computing, precise synchronization between individual quantum processing cores is paramount. Each core's operational state and data transfer are modulated by highly specific frequencies. To maintain coherence and prevent quantum decoherence across a network of cores, a complex calculation involving a base frequency, a synchronization exponent, and a system modulus is performed.

Your task is to implement a function that calculates the modulated frequency. Given three integer values:

  • B (Base Frequency): The initial frequency value of a quantum core.
  • E (Synchronization Exponent): The power to which the base frequency needs to be raised for synchronization.
  • M (System Modulus): A specific modulus value dictated by the quantum network's architecture.

You need to compute (BE) % M. Directly computing BE can result in extremely large numbers that might exceed typical integer data type limits or lead to inefficient computation, especially when 'B' and 'E' are large. Therefore, you must find an efficient way to perform this calculation while applying the modulo operator at appropriate steps.

For example, if B=3, E=4, and M=5:

  • First, calculate 34, which is 81.
  • Then, apply the modulus: 81 % 5, which equals 1.

Input Format

The first line of input contains an integer 'T', denoting the number of synchronization tasks you need to perform.

Following 'T' lines, each contains three space-separated integers: 'B' (base frequency), 'E' (synchronization exponent), and 'M' (system modulus).

Output Format

For each synchronization task, output a single integer on a new line, representing the calculated modulated frequency (BE) % M.

Constraints

  • 1 <= T <= 104
  • 1 <= B, E <= 105
  • 1 <= M <= 104
📌 Constraints
1 <= T <= 10^4; 1 <= x, y <= 10^5; 1 <= z <= 10^4
📝 Sample Input/Output
Sample Input
2
5 2 5
3 3 2
Expected Output
0
1
Explanation for Sample Input
Task 1:
  • Input: B = 5, E = 2, M = 5
  • Calculation: (52) % 5 = 25 % 5 = 0
  • Output: 0
Task 2:
  • Input: B = 3, E = 3, M = 2
  • Calculation: (33) % 2 = 27 % 2 = 1
  • Output: 1
💡 Hints
  • 💡 Directly calculating B^E for large E will lead to integer overflow or extremely slow computation. Remember that the modulo operator interacts well with multiplication: (A * B) % M = ((A % M) * (B % M)) % M.
  • 💡 Consider how the exponent 'E' can be represented in binary. Can you decompose B^E into products of terms like B^1, B^2, B^4, B^8, etc., and apply the modulo at each multiplication step? This approach is known as modular exponentiation or exponentiation by squaring.
  • 💡 Initialize your result to 1. In each step of your calculation, make sure to take the modulo M to keep intermediate results manageable and prevent overflow. Also, update the base by squaring it and taking modulo M.
📚 Topics

Math & Bit Manipulation

🏢 Asked By Companies

Google

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