Simultaneous Congruence Calculator Guide
Core Idea
A simultaneous congruence problem asks for one number that satisfies several remainder conditions. Each condition has a residue and a modulus. The calculator checks every condition before merging them. It also works when moduli are not coprime. That matters because many simple tools only handle clean Chinese remainder theorem cases.
What the Calculator Solves
The tool accepts lists such as residues 2, 3, 2 and moduli 3, 5, 7. This means x leaves remainder 2 after division by 3, remainder 3 after division by 5, and remainder 2 after division by 7. The answer is shown as a least non-negative representative and as a general solution. You may also request solutions inside a custom range.
Why Compatibility Matters
Two congruences can conflict. For example, x is 1 modulo 4 and x is 2 modulo 6 has no solution. The reason is simple. The difference between residues must be divisible by the greatest common divisor of the two moduli. The calculator reports that test during each merge step, so the reason for failure is clear.
Practical Uses
Simultaneous congruences appear in number theory, coding puzzles, calendars, rotations, repeating schedules, and cryptography lessons. They can also model cycles that meet again after different periods. The merged modulus describes the repeat length. The representative shows the first aligned value in that cycle.
Result Interpretation
When a solution exists, the final form is x equals r plus M times t. Here, r is the representative. M is the merged modulus. The integer t can be any whole number. This format describes all possible answers, not just one answer.
Export Features
The CSV option stores the normalized equations, merge steps, and listed range values. The PDF button creates a compact report from the current result. These exports are useful for assignments, records, tutorials, and checking work later.
Best Input Practice
Use positive moduli only. Keep residue and modulus lists the same length. Separate values with commas, spaces, or semicolons. Use labels when you want named equations in the report. Review normalized residues when negative or large residues are entered. Always compare the computed range output with the general form. The displayed list may be shortened when many values qualify in practice.