Random Chinese Remainder Exercise Generator
Build a bounded reproducible chinese remainder exercise from explicitly supplied candidate scenarios. Every candidate must satisfy imported number/chinese-remainder before selection; answer cards preserve its units, conventions, domain limitations and signed/unknown results. Draws are uniform over caller candidate positions; no replacement means no repeated positions, not necessarily distinct answers. No automatic pedagogical correctness, compliance or physical readiness.
Description
The Chinese remainder problem combines congruences into a common integer solution.
The Chinese remainder problem combines congruences into a common integer solution.1
These exercises support consistent non-coprime moduli and return exact decimal strings for the smallest nonnegative solution and period.
When to use Random Chinese Remainder Exercise Generator
Use this tool when you need to assemble reproducible practice cards from a supplied, validated scenario corpus. Prepare a reproducible seed and the inventory, corpus, or experiment parameters stated below.
How to use Random Chinese Remainder Exercise Generator and understand its results
- Prepare a reproducible seed and the inventory, corpus, or experiment parameters stated below.
- Enter seed, count, withoutReplacement, candidateInputsJson, following the field table and calculation rules below, then run the tool.
- Read result. For result objects, inspect the nested output fields below, including any missing evidence or selected indices.
- Correct any input issues and run again. Before using a successful result, review its units, signs, evidence coverage, and stated model or reference assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| seed | Required nonblank deterministic seed; the SDK can replace the source with context.random. | string | Required |
| count | Number of draws, cards, trials, or output entries under the tool-specific rule; safe integer 1–64 with any narrower inventory bound. | number | Required |
| withoutReplacement | True removes selected inventory positions; false permits repeated positions. Equal rendered values or answers can still occur. | boolean | Required |
| candidateInputsJson | Dense list of JSON objects serialized as strings for the imported calculator; every candidate must be valid. | list of string | Required |
Enter 1–128 dense candidateInputsJson strings, each a strict JSON object for the imported calculator. Each string is at most 8192 UTF-16 units, the corpus at most 131072, and nesting at most 16 levels; nonfinite numbers and prototype-control keys are rejected. Select candidate positions under the sampling rules, then validate every candidate, including undrawn ones, by invoking the imported calculator with a fixed random source of 0.5. Any invalid candidate fails the whole request. Invoke it again for selected candidates with the request’s advancing source, and return original inputs, actual answer objects, selected indices, and never-selected indices. Duplicate candidate texts or equal answers may occur at distinct positions.2, 3, 4
Use a nonblank seed of at most 256 UTF-16 units and a safe-integer count from 1 to 64, subject to the tool’s narrower inventory bound. By default the seed is hashed into a deterministic pseudorandom state; each run restarts that state, so identical full inputs and implementation reproduce the result. The SDK may provide context.random, which replaces the seeded source and advances across the request; the seed remains required. Every source value must be finite in [0,1), with a request limit of 100000 source calls. Draw an index as floor(U×remainingSize). For indexed inventory draws, with replacement draw from the full inventory each time; without replacement remove each chosen position from an ordered pool and reject counts exceeding the inventory size. Uniform-source draws are over positions, not necessarily distinct values. This is not secure randomness.
The imported helper requires one to one hundred aligned residue/modulus pairs of safe integers with positive moduli. Merge each congruence using gcd compatibility and an extended-Euclidean inverse, fail on inconsistency, and normalize the solution within the lcm period. BigInt arithmetic preserves the combined result.
x≡r_i (mod m_i); period=lcm(m_i) Read these symbols with the field meanings and branch or reference rules above.
| Output field | Meaning | Value type |
|---|---|---|
| result | Composite output; read its nested fields and the worked example. | object |
| result.baseToolId | Canonical identifier of the imported answer calculator. | string |
| result.selectedCandidateIndices | Zero-based positions in the original candidate corpus, in draw order. | list |
| result.unusedCandidateIndices | Original candidate positions never selected during this request. | list |
| result.exercises | Selected supplied input objects paired with their actual imported answer objects and any stated additional outputs. | list |
Each result.exercises entry preserves its supplied input and imported answer object. Additional roundTripAnswer, secondaryAnswer, prompts, or derived quantities appear only where the mechanics above describe them. selectedCandidateIndices and unusedCandidateIndices refer to the original corpus positions, not answer values.
{
"input": {
"seed": "tool.io/batch42",
"count": 1,
"withoutReplacement": true,
"candidateInputsJson": [
"{\"residues\": [2, 3, 2], \"moduli\": [3, 5, 7]}"
]
},
"output": {
"result": {
"baseToolId": "number/chinese-remainder",
"selectedCandidateIndices": [
0
],
"unusedCandidateIndices": [],
"exercises": [
{
"input": {
"residues": [
2,
3,
2
],
"moduli": [
3,
5,
7
]
},
"answer": {
"solution": "23",
"period": "105"
}
}
]
}
}
}Limitations and assumptions
The default source is deterministic pseudorandomness, not an independently audited or secure draw. Replacement applies to inventory positions; distinct positions may render identically or yield equal answers. A fixture does not independently verify the supplied answers or reference content. Each answer retains the imported calculator’s unit conventions and domain assumptions. Randomizing a supplied scenario does not broaden that calculator’s capabilities.
Missing required fields, unknown fields, wrong types, and violated calculation rules return input issues. Numeric values must be finite; nonfinite numeric outputs are rejected. Arrays must be dense, with no missing cells. Extremely large composite outputs are rejected.
Alternative or Complementary methods
For a related task, use Random Continued Fraction Exercise Generator: A simple continued fraction records successive floor quotients of a rational number. These exercises retain the proper floor behavior for negative numerators rather than truncating division toward zero. For a complementary step, use Random Divisibility Exercise Generator: Integer divisibility asks whether division leaves a zero remainder. These quizzes sample supplied safe-integer pairs and retain signed operands while rejecting a zero divisor.
References
Similar or alternative tools
- Random Continued Fraction Exercise Generator
A simple continued fraction records successive floor quotients of a rational number. These exercises retain the proper floor behavior for negative numerators rather than truncating division toward zero.
- Random Divisibility Exercise Generator
Integer divisibility asks whether division leaves a zero remainder. These quizzes sample supplied safe-integer pairs and retain signed operands while rejecting a zero divisor.