{
  "title": "Modular arithmetic workspace",
  "human_url": "https://happytails.ai/en/calculators/modular-arithmetic-workspace/",
  "agent_url": "https://happytails.ai/agents/en/calculators/modular-arithmetic-workspace/",
  "language": "en",
  "markdown_url": "https://happytails.ai/en/calculators/modular-arithmetic-workspace.md",
  "content": "Modular arithmetic workspace\nSolve congruences, find modular inverses and combine compatible non-coprime systems with worked Euclidean steps.\nHow can we improve this tool?\nReport a problem or share an idea.\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\nHow to use Modular arithmetic workspace\nPaste a JSON example and edit its documented fields. Keep machine field names unchanged.\nChoose a supported operation and review the stated domain and bounds.\nReview the result and download the complete JSON or CSV report, including worked steps.\nUseful next steps\nLinear system solver\nMatrix workspace\nDiophantine equation solver\nModular arithmetic workspace\nChoose linear congruence a*x ≡ b mod modulus, modular inverse, or a system of up to 32 residue/modulus pairs. Moduli must be integers at least two. The extended Euclidean algorithm supplies Bezout coefficients and division steps. A linear congruence exists exactly when gcd(a,modulus) divides b. Compatible non-coprime systems are combined using generalized CRT; incompatible systems report no solution. A residue and positive period parameterize every integer solution without enumerating a huge residue set.\nKeep the complete calculation\nAll processing stays on your device. Paste JSON with dot decimals; quote exact fractions and large integers. Input is limited to 100,000 characters and 100 digits per scalar. Exact results use reduced fractions, with a 1,024-digit and 200,000-step arithmetic bound. Enumeration is limited to 10,000 complete results and 1,000,000 search steps; exceeding a bound returns an error, not a partial answer. Geometry uses floating-point approximations. JSON and typed JSON Pointer CSV retain the complete input, results, constraints and worked steps. Reports are limited to 4,000,000 JSON and 8,000,000 CSV characters. Each calculation uses a disposable worker with a 10-second timeout. Search and sort the first 200 preview rows. Copy result copies the summary. Download JSON or CSV for every result and worked step. Geometry tools also provide a downloadable SVG diagram.",
  "content_format": "text/plain",
  "visibility": "public",
  "links": [
    {
      "name": "Linear system solver",
      "url": "https://happytails.ai/en/calculators/linear-system-solver/"
    },
    {
      "name": "Matrix workspace",
      "url": "https://happytails.ai/en/calculators/matrix-workspace/"
    },
    {
      "name": "Diophantine equation solver",
      "url": "https://happytails.ai/en/calculators/diophantine-equation-solver/"
    }
  ],
  "tool": {
    "id": "modular-arithmetic-workspace",
    "category": "calculators",
    "built": true,
    "requirements": "Choose linear congruence a*x ≡ b mod modulus, modular inverse, or a system of up to 32 residue/modulus pairs. Moduli must be integers at least two. The extended Euclidean algorithm supplies Bezout coefficients and division steps. A linear congruence exists exactly when gcd(a,modulus) divides b. Compatible non-coprime systems are combined using generalized CRT; incompatible systems report no solution. A residue and positive period parameterize every integer solution without enumerating a huge residue set. All processing stays on your device. Paste JSON with dot decimals; quote exact fractions and large integers. Input is limited to 100,000 characters and 100 digits per scalar. Exact results use reduced fractions, with a 1,024-digit and 200,000-step arithmetic bound. Enumeration is limited to 10,000 complete results and 1,000,000 search steps; exceeding a bound returns an error, not a partial answer. Geometry uses floating-point approximations. JSON and typed JSON Pointer CSV retain the complete input, results, constraints and worked steps. Reports are limited to 4,000,000 JSON and 8,000,000 CSV characters. Each calculation uses a disposable worker with a 10-second timeout. Search and sort the first 200 preview rows. Copy result copies the summary. Download JSON or CSV for every result and worked step. Geometry tools also provide a downloadable SVG diagram.",
    "implementation": {
      "fields": [
        {
          "key": "operation",
          "label": "Operation",
          "value": "Linear congruence",
          "type": "select",
          "values": [
            "Linear congruence",
            "Modular inverse",
            "Congruence system"
          ]
        }
      ],
      "presentation": "metrics",
      "inputMaxLength": 100000,
      "inputErrorCode": "EXACT_MATH_INPUT",
      "note": "Choose linear congruence a*x ≡ b mod modulus, modular inverse, or a system of up to 32 residue/modulus pairs. Moduli must be integers at least two. The extended Euclidean algorithm supplies Bezout coefficients and division steps. A linear congruence exists exactly when gcd(a,modulus) divides b. Compatible non-coprime systems are combined using generalized CRT; incompatible systems report no solution. A residue and positive period parameterize every integer solution without enumerating a huge residue set. All processing stays on your device. Paste JSON with dot decimals; quote exact fractions and large integers. Input is limited to 100,000 characters and 100 digits per scalar. Exact results use reduced fractions, with a 1,024-digit and 200,000-step arithmetic bound. Enumeration is limited to 10,000 complete results and 1,000,000 search steps; exceeding a bound returns an error, not a partial answer. Geometry uses floating-point approximations. JSON and typed JSON Pointer CSV retain the complete input, results, constraints and worked steps. Reports are limited to 4,000,000 JSON and 8,000,000 CSV characters. Each calculation uses a disposable worker with a 10-second timeout.",
      "experience": {
        "automatic": false,
        "single": false,
        "inputMaxLength": 100000,
        "inputMode": "text",
        "label": "JSON input",
        "placeholder": "{\n  \"a\": 6,\n  \"b\": 8,\n  \"modulus\": 14\n}",
        "help": "Choose linear congruence a*x ≡ b mod modulus, modular inverse, or a system of up to 32 residue/modulus pairs. Moduli must be integers at least two. The extended Euclidean algorithm supplies Bezout coefficients and division steps. A linear congruence exists exactly when gcd(a,modulus) divides b. Compatible non-coprime systems are combined using generalized CRT; incompatible systems report no solution. A residue and positive period parameterize every integer solution without enumerating a huge residue set. All processing stays on your device. Paste JSON with dot decimals; quote exact fractions and large integers. Input is limited to 100,000 characters and 100 digits per scalar. Exact results use reduced fractions, with a 1,024-digit and 200,000-step arithmetic bound. Enumeration is limited to 10,000 complete results and 1,000,000 search steps; exceeding a bound returns an error, not a partial answer. Geometry uses floating-point approximations. JSON and typed JSON Pointer CSV retain the complete input, results, constraints and worked steps. Reports are limited to 4,000,000 JSON and 8,000,000 CSV characters. Each calculation uses a disposable worker with a 10-second timeout. Up to 100,000 characters. Review the settings before running.",
        "action": null
      },
      "mode": "explicit",
      "interactive": false
    },
    "execution": "Browser interface",
    "api": null,
    "browser_agent": {
      "supported": true,
      "name": "run_current_tool",
      "discovery": "WebMCP on the human page in a supporting browser",
      "verification": "See audit/API-AUDIT.md; availability is not verification"
    }
  },
  "markdown": "# Modular arithmetic workspace\n\n- Human page: https://happytails.ai/en/calculators/modular-arithmetic-workspace/\n- Markdown: https://happytails.ai/en/calculators/modular-arithmetic-workspace.md\n- Structured JSON: https://happytails.ai/agents/en/calculators/modular-arithmetic-workspace/index.json\n- Visibility: public\n\n## Tool capabilities\n\n| Property | Value |\n| --- | --- |\n| Tool ID | `modular-arithmetic-workspace` |\n| Category | calculators |\n| Implementation | Registered implementation. Registration alone is not a production-quality guarantee. |\n| Browser execution | Use the visible action or interactive controls. |\n| Browser processing | Browser; see tool notes for network behavior. |\n| HTTP API | Not callable through HTTP. |\n| Browser-agent interface | run_current_tool |\n\n### Implementation notes\n\nChoose linear congruence a*x ≡ b mod modulus, modular inverse, or a system of up to 32 residue/modulus pairs. Moduli must be integers at least two. The extended Euclidean algorithm supplies Bezout coefficients and division steps. A linear congruence exists exactly when gcd(a,modulus) divides b. Compatible non-coprime systems are combined using generalized CRT; incompatible systems report no solution. A residue and positive period parameterize every integer solution without enumerating a huge residue set. All processing stays on your device. Paste JSON with dot decimals; quote exact fractions and large integers. Input is limited to 100,000 characters and 100 digits per scalar. Exact results use reduced fractions, with a 1,024-digit and 200,000-step arithmetic bound. Enumeration is limited to 10,000 complete results and 1,000,000 search steps; exceeding a bound returns an error, not a partial answer. Geometry uses floating-point approximations. JSON and typed JSON Pointer CSV retain the complete input, results, constraints and worked steps. Reports are limited to 4,000,000 JSON and 8,000,000 CSV characters. Each calculation uses a disposable worker with a 10-second timeout.\n\n### Inputs and settings\n\nText input: JSON input.\n\nExample or placeholder shown in the interface (not a validated fixture):\n\n```text\n{\n  \"a\": 6,\n  \"b\": 8,\n  \"modulus\": 14\n}\n```\n\nChoose linear congruence a*x ≡ b mod modulus, modular inverse, or a system of up to 32 residue/modulus pairs. Moduli must be integers at least two. The extended Euclidean algorithm supplies Bezout coefficients and division steps. A linear congruence exists exactly when gcd(a,modulus) divides b. Compatible non-coprime systems are combined using generalized CRT; incompatible systems report no solution. A residue and positive period parameterize every integer solution without enumerating a huge residue set. All processing stays on your device. Paste JSON with dot decimals; quote exact fractions and large integers. Input is limited to 100,000 characters and 100 digits per scalar. Exact results use reduced fractions, with a 1,024-digit and 200,000-step arithmetic bound. Enumeration is limited to 10,000 complete results and 1,000,000 search steps; exceeding a bound returns an error, not a partial answer. Geometry uses floating-point approximations. JSON and typed JSON Pointer CSV retain the complete input, results, constraints and worked steps. Reports are limited to 4,000,000 JSON and 8,000,000 CSV characters. Each calculation uses a disposable worker with a 10-second timeout. Up to 100,000 characters. Review the settings before running.\n\n| Setting | Meaning | Control type | Default | Constraints |\n| --- | --- | --- | --- | --- |\n| `operation` | Operation | select | \"Linear congruence\" | Allowed: Linear congruence, Modular inverse, Congruence system |\n\n### Browser-agent workflow\n\n1. Open the human page in a browser that supports WebMCP.\n2. Discover the interface exposed by that page; use its actual schema.\n3. Select files on the page first when the tool requires files.\n4. Call `run_current_tool` with valid input and settings.\n5. Inspect the returned result or error and the displayed output. Retrieve files from the displayed download links.\n\nCalls do not automatically copy or download results. Browser permissions still apply. Interface availability alone is not proof of successful execution.\n\n### Planning specification\n\nChoose linear congruence a*x ≡ b mod modulus, modular inverse, or a system of up to 32 residue/modulus pairs. Moduli must be integers at least two. The extended Euclidean algorithm supplies Bezout coefficients and division steps. A linear congruence exists exactly when gcd(a,modulus) divides b. Compatible non-coprime systems are combined using generalized CRT; incompatible systems report no solution. A residue and positive period parameterize every integer solution without enumerating a huge residue set. All processing stays on your device. Paste JSON with dot decimals; quote exact fractions and large integers. Input is limited to 100,000 characters and 100 digits per scalar. Exact results use reduced fractions, with a 1,024-digit and 200,000-step arithmetic bound. Enumeration is limited to 10,000 complete results and 1,000,000 search steps; exceeding a bound returns an error, not a partial answer. Geometry uses floating-point approximations. JSON and typed JSON Pointer CSV retain the complete input, results, constraints and worked steps. Reports are limited to 4,000,000 JSON and 8,000,000 CSV characters. Each calculation uses a disposable worker with a 10-second timeout. Search and sort the first 200 preview rows. Copy result copies the summary. Download JSON or CSV for every result and worked step. Geometry tools also provide a downloadable SVG diagram.\n\nThis describes intended requirements. Use the implementation notes, interface schema and observed output to determine current support.\n\n### Complete machine-readable capability record\n\n```json\n{\n  \"id\": \"modular-arithmetic-workspace\",\n  \"category\": \"calculators\",\n  \"built\": true,\n  \"requirements\": \"Choose linear congruence a*x ≡ b mod modulus, modular inverse, or a system of up to 32 residue/modulus pairs. Moduli must be integers at least two. The extended Euclidean algorithm supplies Bezout coefficients and division steps. A linear congruence exists exactly when gcd(a,modulus) divides b. Compatible non-coprime systems are combined using generalized CRT; incompatible systems report no solution. A residue and positive period parameterize every integer solution without enumerating a huge residue set. All processing stays on your device. Paste JSON with dot decimals; quote exact fractions and large integers. Input is limited to 100,000 characters and 100 digits per scalar. Exact results use reduced fractions, with a 1,024-digit and 200,000-step arithmetic bound. Enumeration is limited to 10,000 complete results and 1,000,000 search steps; exceeding a bound returns an error, not a partial answer. Geometry uses floating-point approximations. JSON and typed JSON Pointer CSV retain the complete input, results, constraints and worked steps. Reports are limited to 4,000,000 JSON and 8,000,000 CSV characters. Each calculation uses a disposable worker with a 10-second timeout. Search and sort the first 200 preview rows. Copy result copies the summary. Download JSON or CSV for every result and worked step. Geometry tools also provide a downloadable SVG diagram.\",\n  \"implementation\": {\n    \"fields\": [\n      {\n        \"key\": \"operation\",\n        \"label\": \"Operation\",\n        \"value\": \"Linear congruence\",\n        \"type\": \"select\",\n        \"values\": [\n          \"Linear congruence\",\n          \"Modular inverse\",\n          \"Congruence system\"\n        ]\n      }\n    ],\n    \"presentation\": \"metrics\",\n    \"inputMaxLength\": 100000,\n    \"inputErrorCode\": \"EXACT_MATH_INPUT\",\n    \"note\": \"Choose linear congruence a*x ≡ b mod modulus, modular inverse, or a system of up to 32 residue/modulus pairs. Moduli must be integers at least two. The extended Euclidean algorithm supplies Bezout coefficients and division steps. A linear congruence exists exactly when gcd(a,modulus) divides b. Compatible non-coprime systems are combined using generalized CRT; incompatible systems report no solution. A residue and positive period parameterize every integer solution without enumerating a huge residue set. All processing stays on your device. Paste JSON with dot decimals; quote exact fractions and large integers. Input is limited to 100,000 characters and 100 digits per scalar. Exact results use reduced fractions, with a 1,024-digit and 200,000-step arithmetic bound. Enumeration is limited to 10,000 complete results and 1,000,000 search steps; exceeding a bound returns an error, not a partial answer. Geometry uses floating-point approximations. JSON and typed JSON Pointer CSV retain the complete input, results, constraints and worked steps. Reports are limited to 4,000,000 JSON and 8,000,000 CSV characters. Each calculation uses a disposable worker with a 10-second timeout.\",\n    \"experience\": {\n      \"automatic\": false,\n      \"single\": false,\n      \"inputMaxLength\": 100000,\n      \"inputMode\": \"text\",\n      \"label\": \"JSON input\",\n      \"placeholder\": \"{\\n  \\\"a\\\": 6,\\n  \\\"b\\\": 8,\\n  \\\"modulus\\\": 14\\n}\",\n      \"help\": \"Choose linear congruence a*x ≡ b mod modulus, modular inverse, or a system of up to 32 residue/modulus pairs. Moduli must be integers at least two. The extended Euclidean algorithm supplies Bezout coefficients and division steps. A linear congruence exists exactly when gcd(a,modulus) divides b. Compatible non-coprime systems are combined using generalized CRT; incompatible systems report no solution. A residue and positive period parameterize every integer solution without enumerating a huge residue set. All processing stays on your device. Paste JSON with dot decimals; quote exact fractions and large integers. Input is limited to 100,000 characters and 100 digits per scalar. Exact results use reduced fractions, with a 1,024-digit and 200,000-step arithmetic bound. Enumeration is limited to 10,000 complete results and 1,000,000 search steps; exceeding a bound returns an error, not a partial answer. Geometry uses floating-point approximations. JSON and typed JSON Pointer CSV retain the complete input, results, constraints and worked steps. Reports are limited to 4,000,000 JSON and 8,000,000 CSV characters. Each calculation uses a disposable worker with a 10-second timeout. Up to 100,000 characters. Review the settings before running.\",\n      \"action\": null\n    },\n    \"mode\": \"explicit\",\n    \"interactive\": false\n  },\n  \"execution\": \"Browser interface\",\n  \"api\": null,\n  \"browser_agent\": {\n    \"supported\": true,\n    \"name\": \"run_current_tool\",\n    \"discovery\": \"WebMCP on the human page in a supporting browser\",\n    \"verification\": \"See audit/API-AUDIT.md; availability is not verification\"\n  }\n}\n```\n\n## Page guide\n\nSolve congruences, find modular inverses and combine compatible non-coprime systems with worked Euclidean steps.\n\nHow can we improve this tool?\n\nReport a problem or share an idea.\n\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\n\n### How to use Modular arithmetic workspace\n\n1. Paste a JSON example and edit its documented fields. Keep machine field names unchanged.\n2. Choose a supported operation and review the stated domain and bounds.\n3. Review the result and download the complete JSON or CSV report, including worked steps.\n\n## Useful next steps\n\n- [Linear system solver](https://happytails.ai/en/calculators/linear-system-solver/) ([Markdown](https://happytails.ai/en/calculators/linear-system-solver.md))\n\n- [Matrix workspace](https://happytails.ai/en/calculators/matrix-workspace/) ([Markdown](https://happytails.ai/en/calculators/matrix-workspace.md))\n\n- [Diophantine equation solver](https://happytails.ai/en/calculators/diophantine-equation-solver/) ([Markdown](https://happytails.ai/en/calculators/diophantine-equation-solver.md))\n\n## Modular arithmetic workspace\n\nChoose linear congruence a\\*x ≡ b mod modulus, modular inverse, or a system of up to 32 residue/modulus pairs. Moduli must be integers at least two. The extended Euclidean algorithm supplies Bezout coefficients and division steps. A linear congruence exists exactly when gcd(a,modulus) divides b. Compatible non-coprime systems are combined using generalized CRT; incompatible systems report no solution. A residue and positive period parameterize every integer solution without enumerating a huge residue set.\n\n### Keep the complete calculation\n\nAll processing stays on your device. Paste JSON with dot decimals; quote exact fractions and large integers. Input is limited to 100,000 characters and 100 digits per scalar. Exact results use reduced fractions, with a 1,024-digit and 200,000-step arithmetic bound. Enumeration is limited to 10,000 complete results and 1,000,000 search steps; exceeding a bound returns an error, not a partial answer. Geometry uses floating-point approximations. JSON and typed JSON Pointer CSV retain the complete input, results, constraints and worked steps. Reports are limited to 4,000,000 JSON and 8,000,000 CSV characters. Each calculation uses a disposable worker with a 10-second timeout. Search and sort the first 200 preview rows. Copy result copies the summary. Download JSON or CSV for every result and worked step. Geometry tools also provide a downloadable SVG diagram.\n"
}