This theorem tells us that congruences modulo $m$ behave nicely with respect to addition, subtraction, and multiplication. In other words, you can perform these arithmetic operations within the modular system without affecting the congruence.

Let's break down what each part means and why it's important:

1. a′+b′≡a+b(mod$m$): If a' is congruent to a (mod m) and b' is congruent to b (mod m), then the sum of a' and b' is congruent to the sum of a and b (mod m).

   Example: Let m = 5. If a = 7 and a' = 12 (both congruent to 2 mod 5), and b = 3 and b' = 8 (both congruent to 3 mod 5), then a + b = 10 and a' + b' = 20. Both 10 and 20 are congruent to 0 (mod 5).

2. a′−b′≡a−b(mod$m$): Similar to addition, if a' is congruent to a (mod m) and b' is congruent to b (mod m), then the difference of a' and b' is congruent to the difference of a and b (mod m).

   Example: Let m = 5. Using the same values as above, a - b = 4 and a' - b' = 4. Both are congruent to 4 (mod 5). Note that even if the subtraction results in a negative number, the congruence still holds (you might need to add a multiple of m to get a positive representative).

3. a′b′≡ab(mod$m$): This is perhaps the most important part. If a' is congruent to a (mod m) and b' is congruent to b (mod m), then the product of a' and b' is congruent to the product of a and b (mod m).

   Example: Let m = 5. Again, using the same values, a * b = 21 and a' * b' = 96. Both 21 and 96 are congruent to 1 (mod 5).

Why is this theorem important?

This theorem is fundamental to modular arithmetic. It allows us to perform calculations with congruences as if they were regular equations (with some caveats). This is crucial in areas like:

• Cryptography: Many cryptographic algorithms rely heavily on modular arithmetic.
• Computer Science: Hashing functions and other data structures use modular arithmetic.
• Number Theory: This theorem is a building block for more advanced number theory concepts.

Essentially, it simplifies working with congruences by letting us substitute congruent numbers without changing the final result (modulo m).

🧠Demonstration of Theorem

https://gist.github.com/viadean/fd1b426e9bf652ec6dc3e1aece7613dc

Explanation and Key Improvements:

1. Clear Function Definitions: The code is organized into functions for each part of the theorem (addition, subtraction, multiplication). This makes it more readable and reusable.
2. Explicit Modulo Operator: The % operator (modulo) is used consistently to ensure that all calculations are done within the modulo m system. This is crucial for correctness.
3. Condition Check: The code now explicitly checks the initial condition: a_prime % m == a % m and b_prime % m == b % m. If this condition isn't met, the functions print a message and return False, indicating that the theorem's premise isn't satisfied. This is very important for a proper "proof" by code.
4. Informative Output: The print statements clearly show the calculations and the result of the congruence check, making it easy to follow what's happening.
5. Return Values: The functions return True if the congruence holds and False otherwise. This allows you to use the functions in more complex logic if needed.
6. Handles Negative Results: The subtraction example shows how the modulo operator works with negative numbers in Python, ensuring the congruence is still checked correctly. 3 % 7 in Python evaluates to 4, so the congruence is considered true.