Random Linear Congruence Practice Set Generator

Description

Generate bounded linear congruences with exact delegated residue/period keys and enumerate every solution class modulo the supplied generated modulus.

A generated practice exercise contains a prompt and an answer key under a specific rule set. A planted answer establishes one possible solution; only an explicit enumeration or uniqueness check establishes whether alternatives exist.

Generate bounded linear congruences with exact delegated residue/period keys and enumerate every solution class modulo the supplied generated modulus.

When to use this tool

Use this generator to create reproducible linear congruence practice set cases from the supplied model or bank. Inspect the returned evidence before using cases in a test or exercise.

How to use the tool and how it works

Supply count, magnitude, maximumModulus in the field shapes below. Set seed when you need an explicit replay.
Run the tool and correct any input error. For generated controls, distinguish a successful fixture containing an invalid case from failure to generate the fixture.
Read cards. Use the conventions below and keep answer-bearing fields separate when presenting an exercise.

Draw signed coefficients and a modulus, delegate exact solution of a*x congruent to b, and expand the returned residue, period, and multiplicity into representatives within one modulus.^[1]

Delegated operations used for this result: linear congruence solution. Their calculation or parsing conventions apply to the corresponding result fields.^[2]

Input field	Meaning and units	Accepted input
count	Requested count; the mechanics explain whether one draw expands into several returned records.	Required. Safe integer; minimum 1; maximum 16.
magnitude	Magnitude	Required. Safe integer; minimum 0; maximum 1000000.
maximumModulus	Maximum Modulus	Required. Safe integer; minimum 2; maximum 64.
seed	Optional numeric seed; takes precedence over context randomness. Without either, use the fixed tool.io default seed.	Optional. Safe integer.

Output field	Interpretation	Returned shape
cards	Cards	Required. Array of Object; no unknown keys.; minimum items 1; maximum items 16.
cards[].a	A	Required. Safe integer.
cards[].b	B	Required. Safe integer.
cards[].modulus	Modulus	Required. Safe integer.
cards[].solvable	Solvable	Required. Boolean.
cards[].residue	Residue	Required. Text.
cards[].period	Period	Required. Text.
cards[].multiplicity	Multiplicity	Required. Text.
cards[].solutions	Solutions	Required. Array of Safe integer.; minimum items 0; maximum items 64.
randomness	Replay metadata: source is seed or context; seed is the normalized 32-bit seed or null for a context source.	Required. Object; no unknown keys.
randomness.seed	Optional numeric replay seed; normalization and precedence are explained below.	Required. Safe integer. or Null.
randomness.source	Source	Required. Text; one of "context", "seed".

An explicit safe-integer seed takes precedence over injected context randomness and is normalized to an unsigned 32-bit integer. With no seed and no context source, replay uses the fixed tool.io default seed. A context source must return finite values from zero inclusive to one exclusive; its advancing state must be retained separately. randomness records the source and normalized seed, or null for context. These generators are not cryptographic random sources.

json

{
  "input": {
    "count": 1,
    "magnitude": 3,
    "maximumModulus": 8,
    "seed": 42
  },
  "output": {
    "cards": [
      {
        "a": 1,
        "b": 0,
        "modulus": 7,
        "solvable": true,
        "residue": "0",
        "period": "7",
        "multiplicity": "1",
        "solutions": [
          0
        ]
      }
    ],
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

An unsolvable congruence has an empty solutions list. Exact calculator fields use decimal strings while generated coefficients and displayed representatives are bounded safe integers.
A replay requires the same complete input and implementation. Randomized construction does not guarantee uniform coverage of every feasible output; finite samples can repeat unless a distinct-selection rule is explicitly stated.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

For a supplied example, inspect the returned construction or delegated check directly. To evaluate a consumer, also test cases chosen for its own boundaries; a generated finite corpus does not establish exhaustive coverage.

References

Tool behavior and example: Random Linear Congruence Practice Set Generator — tool.io
Calculation or validation: math/linear-congruence-solution — tool.io

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