Solving Modular Arithmetic Problems: A Case Study on Dividing Numbers by 121 and 11

Modular arithmetic, often used in computer science and cryptography, involves the division of numbers and finding remainders. This article explores a specific problem and provides a detailed solution, demonstrating the step-by-step process of calculating remainders using Python.

Understanding the Problem

The problem at hand is as follows: If a number, let's denote it as ( N ), when divided by 121 leaves a remainder of 25, what remainder will it leave when divided by 11?

Mathematical Expressions

Given: N 121k - 25

Where ( k ) is an integer.

Our goal is to find the remainder when ( N ) is divided by 11.

Solving the Problem

We will use the properties of modular arithmetic to solve this. Let's simplify the expression by breaking it down into smaller parts.

Step 1: Modulo 11 Calculations

Step 1.1: Calculate 121 mod 11

121 % 11

Result: 0

Explanation: Since 121 is exactly divisible by 11, ( 121 mod 11 0 ).

Step 1.2: Calculate 25 mod 11

25 % 11

Result: 3

Explanation: 25 divided by 11 gives a quotient of 2 and a remainder of 3, so ( 25 mod 11 3 ).

Step 2: Substitution and Simplification

Substituting the results back into the original equation:

(121 * k - 25) % 11

Simplifies to:

0 * k - 25 % 11

Which further simplifies to:

-25 % 11

Since we already know ( 25 mod 11 3 ), the final expression is:

3

Thus, the remainder when ( N ) is divided by 11 is 3.

Python Code Implementation

def find_remainder(N, divisor1121, divisor211):
    k  (N   25) // divisor1  # Determine k based on the given remainder of 25
    remainder  (divisor1 * k - 25) % divisor2
    return remainder
# Example usage
N  121 * 6   25
print(find_remainder(N))  # Output: 3

Additional Notes

Here are a few additional notes and shortcuts that can be useful:

Note 1: Simplifying the Expression

When given ( N equiv a mod b ) and ( b c^2 ), we can simplify the problem by calculating ( a mod c ).

Note 2: Shortcut

A common shortcut is to divide the remainder directly by the new divisor:

25 % 11

This directly gives the remainder 3.

Conclusion

The remainder when a number that leaves a remainder of 25 when divided by 121 is divided by 11 is 3. This solution is derived using the principles of modular arithmetic and simplification techniques.