Refactorable (TAU) Number Analyzer

This project is an algorithmic implementation in C designed to identify Refactorable Numbers (also known as TAU numbers) within the field of Number Theory.

🧮 Mathematical Definition

A natural number $n$ is called a Refactorable Number or TAU Number if it is divisible by the count of its divisors.

Formally, if $d(n)$ denotes the number of divisors of $n$, then $n$ is a TAU number if:

$$n \equiv 0 \pmod{d(n)}$$

Example Analysis

• Number: 12

• Divisors: 1, 2, 3, 4, 6, 12

• Count of Divisors ($d(n)$): 6

• Check: $12 \div 6 = 2$ (Remainder 0) -> ✅ TAU Number

• Number: 15

• Divisors: 1, 3, 5, 15

• Count of Divisors ($d(n)$): 4

• Check: $15 \div 4 = 3.75$ (Remainder exists) -> ❌ Not a TAU Number

⚙️ How It Works

The algorithm follows a computational approach to factorization:

1. Input Acquisition: Takes an integer input from the user.
2. Iteration & Factorization: Loops through all integers from 1 to $n$ to identify factors.
3. Divisor Counting: Accumulates the total count of positive divisors.
4. Modular Validation: Applies the modulus operator (%) to check if the original number is divisible by the divisor count.
5. Result Output: Displays detailed factorization steps and the final classification.

🚀 Usage

1. Compile the code:

   gcc tau_analyzer.c -o analyzer

2. Run the program:

   ./analyzer

3. Enter a number to verify its property.

This repository demonstrates the application of control structures (loops/conditionals) and arithmetic algorithms in C.