Description
Generate positive integer pairs and a complete Euclidean division trace with one hidden quotient/remainder prompt and a verified gcd key.
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 positive integer pairs and a complete Euclidean division trace with one hidden quotient/remainder prompt and a verified gcd key.
When to use this tool
Use this generator to create reproducible euclidean algorithm trace puzzle 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 maximum 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 first, second, hiddenIndex, gcd, trace, prompt. Use the conventions below. Keep answer-bearing fields separate when presenting an exercise.
Draw positive integers and repeatedly delegate quotient/remainder division until the remainder is zero. Cross-check the final gcd with a separate calculator, then hide one trace row's quotient and remainder as null in the prompt.1
Delegated operations used for this result: divmod; greatest common factor. Their calculation or parsing conventions apply to the corresponding result fields.2, 3
| Input field | Meaning and units | Accepted input |
|---|---|---|
| maximum | Maximum | Required. Safe integer; minimum 2; maximum 1000000. |
| 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 |
|---|---|---|
| first | First | Required. Safe integer. |
| second | Second | Required. Safe integer. |
| hiddenIndex | Hidden Index | Required. Safe integer. |
| gcd | Gcd | Required. Safe integer. |
| trace | Trace | Required. Array of Object; no unknown keys.; minimum items 1; maximum items 32. |
| trace[].dividend | Dividend | Required. Safe integer. |
| trace[].divisor | Divisor | Required. Safe integer. |
| trace[].quotient | Quotient | Required. Safe integer. |
| trace[].remainder | Remainder | Required. Safe integer. |
| prompt | Prompt | Required. Array of Object; no unknown keys.; minimum items 1; maximum items 32. |
| prompt[].dividend | Dividend | Required. Safe integer. |
| prompt[].divisor | Divisor | Required. Safe integer. |
| prompt[].quotient | Quotient | Required. Safe integer. or Null. |
| prompt[].remainder | Remainder | Required. Safe integer. or Null. |
| randomness | Replay source and normalized 32-bit seed, or null seed for an advancing context source. | Required. Object; no unknown keys. |
| randomness.seed | Optional safe-integer 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.
{
"input": {
"maximum": 30,
"seed": 42
},
"output": {
"first": 19,
"second": 14,
"hiddenIndex": 3,
"gcd": 1,
"trace": [
{
"dividend": 19,
"divisor": 14,
"quotient": 1,
"remainder": 5
},
{
"dividend": 14,
"divisor": 5,
"quotient": 2,
"remainder": 4
},
{
"dividend": 5,
"divisor": 4,
"quotient": 1,
"remainder": 1
},
{
"dividend": 4,
"divisor": 1,
"quotient": 4,
"remainder": 0
}
],
"prompt": [
{
"dividend": 19,
"divisor": 14,
"quotient": 1,
"remainder": 5
},
{
"dividend": 14,
"divisor": 5,
"quotient": 2,
"remainder": 4
},
{
"dividend": 5,
"divisor": 4,
"quotient": 1,
"remainder": 1
},
{
"dividend": 4,
"divisor": 1,
"quotient": null,
"remainder": null
}
],
"randomness": {
"seed": 42,
"source": "seed"
}
}
}Limitations and assumptions
- The full trace and gcd reveal the answer and must be separated for practice. This uses bounded positive safe integers rather than arbitrary precision inputs or a proof of performance for unbounded integers.
- A replay requires the same complete input and implementation. The stated sampling convention applies within its eligible space. A finite draw does not exhaust all possible outputs; repeats remain possible unless a distinct-selection rule is explicit.
- 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
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