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Mathematics Final Year Topic: Solvability and Construction of Simultaneous Congruences with Non-Coprime Moduli

This Mathematics final year project characterises consistency conditions for a bounded system of integer congruences.

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Why choose this project topic?

A focused study of generalised congruences gives you a specific question in elementary number theory. It characterises consistency conditions for a bounded system of integer congruences. The bounded comparison creates room to explain how your evidence supports an interpretation and where the method has limits.

When does a simultaneous congruence system with non-coprime moduli have a solution, and how can all solutions be represented?

Agree the apparatus or dataset, comparison range and feasible measurement schedule for generalised congruences with your supervisor.

Proposed project objectives

  1. 01Define the materials, variables and comparison conditions for generalised congruences.
  2. 02Derive the compatibility condition using greatest common divisors.
  3. 03Evaluate the measurements or model outputs in relation to this question: When does a simultaneous congruence system with non-coprime moduli have a solution, and how can all solutions be represented?

A suggested research approach

Derive the compatibility condition using greatest common divisors. Construct solutions with an extended-Euclidean procedure, verify them directly and contrast solvable and inconsistent systems using exact integer arithmetic. Agree the available resources and record uncertainty, deviations from the protocol and any observations that challenge the initial interpretation.

What you will need

  • A specified family of integer congruence systems
  • References on divisibility and the Chinese remainder theorem
  • Exact integer calculations and verification examples

Keep your project scope clear

The coprime Chinese-remainder formula cannot be applied unchanged when compatibility conditions fail.

Mathematics project chapter outline

Use this outline as a starting point. You can edit the chapter titles to match your department’s format during setup.

  1. Chapter 1Introduction
  2. Chapter 2Literature Review
  3. Chapter 3Theory and Methodology
  4. Chapter 4Results and Applications
  5. Chapter 5Summary, Conclusion and Recommendations

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