Calculator Input
Example Data Table
| Case | Equations | Expected interpretation |
|---|---|---|
| Single variable CRT | x ≡ 2 (mod 3), x ≡ 3 (mod 5) | x ≡ 8 (mod 15) |
| Linear congruence | 4x ≡ 8 (mod 12) | x ≡ 2 (mod 3) |
| Two variable system | 2x + 3y ≡ 1 (mod 5), x − y ≡ 2 (mod 7) | Residue tuple search |
| Inconsistent case | 2x ≡ 1 (mod 4) | No solution because gcd(2,4) does not divide 1 |
Formula Used
General equation: a1x + a2y + a3z ≡ r (mod m).
Single variable reduction: ax ≡ r (mod m). Let g = gcd(a,m). A solution exists only when g divides r.
Reduced equation: (a/g)x ≡ (r/g) (mod m/g). The inverse of a/g gives x ≡ (r/g)(a/g)−1 (mod m/g).
CRT compatibility: x ≡ p (mod m) and x ≡ q (mod n) combine only when p − q is divisible by gcd(m,n).
Multi-variable check: A tuple is valid when every left side minus its remainder equals 0 under the related modulus.
How to Use This Calculator
- Choose one, two, or three variables.
- Select the number of modular equations.
- Enter integer coefficients, remainders, and positive moduli.
- Use zero for any coefficient that should not appear.
- Set the residue cap for multi-variable searches.
- Press Calculate System to view results above the form.
- Download the CSV for spreadsheet work or the PDF for reporting.
Article: Modular Equation Systems
Why modular systems matter
A modular system asks for values that satisfy several congruences at once. These problems appear in algebra, cryptography, coding theory, clocks, calendars, and computer science. They are also useful when a repeated cycle hides the real value. A normal equation gives one direct number. A congruence gives a residue class. That class repeats after its modulus.
What this calculator solves
This calculator handles one variable equations with an exact generalized Chinese remainder method. It also handles two and three variable linear congruence systems through a controlled residue search. Each equation can have its own modulus. Coefficients may be positive, zero, or negative. Remainders are normalized when the option is active. The tool then checks every displayed answer against every equation.
Reading the results
A result like x congruent to 4 modulo 15 means every number 4 plus 15k works. Here k is any integer. For more variables, the table lists residue tuples. A tuple such as x equals 2 and y equals 5 means those residues satisfy all active equations within the searched period. When the searched period equals the least common multiple of all moduli, the result is exact for that residue space.
Using advanced options
The search cap protects the page from very large grids. Increase it when moduli are small and you need more residue classes. Lower it when three variables create too many combinations. The solution limit controls how many matching tuples appear. A system can have no solution, one residue class, or many residue tuples. The verification table shows why each result is accepted.
Good input practice
Use integer coefficients and positive integer moduli. Keep moduli reasonable for multi variable searches. For exact one variable work, enter each congruence as a times x is congruent to r modulo m. For systems with two or three variables, fill unused coefficients with zero. Review the formulas before exporting results. Save the CSV for spreadsheets. Use the PDF button for a quick report.
Common interpretation
Always remember that a modular answer describes a repeating family, not only one integer. Test one member, then add the modulus period to see matching values continue across cycle.
FAQs
What is a modular equation system?
It is a group of congruences that must be true at the same time. Each equation uses a modulus, so answers repeat in residue classes.
Can this calculator solve one variable systems exactly?
Yes. One variable systems use linear congruence reduction and a generalized Chinese remainder check. The final answer is a residue class.
How are two and three variable systems solved?
They are solved by checking residue tuples inside the selected search range. A tuple is listed only when every congruence is satisfied.
Why can a modular system have no solution?
A conflict happens when congruences demand incompatible residues. For ax ≡ r mod m, no solution exists when gcd(a,m) does not divide r.
What does the residue search cap mean?
It limits how many residues are tested for each variable. Higher caps search more values but can make large systems slower.
What does x ≡ 4 mod 15 mean?
It means x can be 4 plus any multiple of 15. Values such as 4, 19, 34, and −11 belong to the same class.
Can coefficients be negative?
Yes. Negative coefficients and remainders are accepted. The calculator can normalize remainders into standard nonnegative residues.
What exports are available?
You can download a CSV file for spreadsheet analysis. You can also create a PDF report directly from the displayed result section.