{ "category": "Math and trigonometry functions", "display_name": "GCD", "entry_kind": "built_in_function", "eval_api": { "method": "POST", "url": "https://handbook.dnacalc.com/api/eval" }, "function_id": "FUNC.GCD", "implementation_state": { "answer_bank": { "excel_builds": [ "20430" ], "larger_numeric": 0, "last_bit": 0, "matched": 7966, "other_mismatch": 0, "refused": 0, "rows": 7966, "unread": 0, "worst_ulp": 0 }, "detail": "Recorded: 7,966 in the answer bank and 6 workbook cells.", "dnacalc_commit": "9e1884369a678158dd82d3d74188daf986995e87", "in_catalogue": true, "state": "matches", "state_label": "Matches every recorded Excel answer", "summary": "Matches all 7,972 recorded answers", "workbook_corpus": { "cells": 6, "matched": 6 } }, "implementations": [ { "commit": "9e1884369a678158dd82d3d74188daf986995e87", "flavour": "excel-compatibility-candidate", "language": "Rust", "name": "CalcFunc", "own_code": [ "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/math.rs#L134-L137", "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/ported/gcd_fn.rs#L4-L6" ], "repository": "https://github.com/DnaCalc/DnaCalc", "shared_code": [ "catalogue/math::integer_groups", "ported/gcd_lcm_common::gcd_int" ], "source": "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/math.rs#L134-L137", "state": "matches", "suite": null, "summary": "Matches all 7,972 recorded answers" } ], "json_url": "https://handbook.dnacalc.com/api/functions/FUNC.GCD.json", "name": "GCD", "notes": [ { "basis": "All recorded rows match after the boundary change: GCD 7,966 of 7,966 and LCM 7,967 of 7,967.", "documentation": { "claim": "If a parameter to GCD is >=2^53, GCD returns the #NUM! error value. If LCM(a,b) >=2^53, LCM returns the #NUM! error value.", "source": "https://support.microsoft.com/en-us/office/gcd-function-d5107a51-69e3-461f-8e4c-ddfc21b5073a; https://support.microsoft.com/en-us/office/lcm-function-7152b67a-8bb5-4075-ae5c-06ede5563c94" }, "evidence": [ { "commit": "9c23dde1dfd6d93e742d763a4bad3eaec229b29c", "locator": "recorded_gcd_lcm_admit_the_exact_operand_boundary", "note": "recorded readings pinned", "path": "apps/dna-onecalc/tests/acceptance.rs", "repository": "DnaCalc" } ], "functions": [ "FUNC.GCD", "FUNC.LCM" ], "inferred": null, "kind": "documentation-differs", "note_id": "FN-GCD-001", "open": [], "recorded_on": "2026-11-04", "repros": [ { "call": "GCD(9007199254740992, 1)", "correct": null, "excel": { "bits": "0x3ff0000000000000", "display": "1", "kind": "number" }, "input_route": "cell-value2", "inputs": [ { "name": "number1", "value": { "bits": "0x4340000000000000", "display": "9007199254740992", "kind": "number" } }, { "name": "number2", "value": { "bits": "0x3ff0000000000000", "display": "1", "kind": "number" } } ], "recorded_in": "Excel 16.0 build 20430, Windows 64-bit, CPU not recorded in this capture", "source": { "commit": "4d5320dfdc385354469b4d506153357bb932ab08", "locator": "w111boundary-GCD-00190", "note": null, "path": "runs/w111-integer-publication-20260929/answers/answers-gcd.json.gz", "repository": "CalcFunc-Evidence" } }, { "call": "LCM(9007199254740992, 2)", "correct": null, "excel": { "bits": "0x4340000000000000", "display": "9007199254740992", "kind": "number" }, "input_route": "cell-value2", "inputs": [ { "name": "number1", "value": { "bits": "0x4340000000000000", "display": "9007199254740992", "kind": "number" } }, { "name": "number2", "value": { "bits": "0x4000000000000000", "display": "2", "kind": "number" } } ], "recorded_in": "Excel 16.0 build 20430, Windows 64-bit, CPU not recorded in this capture", "source": { "commit": "4d5320dfdc385354469b4d506153357bb932ab08", "locator": "w111boundary-LCM-00194", "note": null, "path": "runs/w111-integer-publication-20260929/answers/answers-lcm.json.gz", "repository": "CalcFunc-Evidence" } }, { "call": "GCD(9007199254740994, 0)", "correct": null, "excel": { "bits": null, "display": "#NUM!", "kind": "error" }, "input_route": "cell-value2", "inputs": [ { "name": "number1", "value": { "bits": "0x4340000000000001", "display": "9007199254740994", "kind": "number" } }, { "name": "number2", "value": { "bits": "0x0000000000000000", "display": "0", "kind": "number" } } ], "recorded_in": "Excel 16.0 build 20430, Windows 64-bit, CPU not recorded in this capture", "source": { "commit": "4d5320dfdc385354469b4d506153357bb932ab08", "locator": "w111boundary-GCD-00217", "note": null, "path": "runs/w111-integer-publication-20260929/answers/answers-gcd.json.gz", "repository": "CalcFunc-Evidence" } } ], "schema": "efh.function-note/v1", "statement": "GCD accepts an operand equal to 2^53, and LCM accepts an operand or a result equal to 2^53. Only values above 2^53 give #NUM!.", "status": "measured", "title": "GCD and LCM accept 2^53 itself, not only values below it" } ], "schema": "efh.function-reference/v1", "section_order": [ "what-it-does", "try-it", "quirks", "how-to-implement", "version-differences", "alternatives" ], "sections": { "alternatives": "\n### Relationships\n\n- **[MOD](FUNC.MOD.md)** — the primitive the Euclidean algorithm is built from. `GCD` is what\n repeated `MOD` converges to.\n- **[QUOTIENT](FUNC.QUOTIENT.md)** and **[INT](FUNC.INT.md)** — the other truncating integer\n operations; each truncates by a different rule, and only `INT` floors.\n- **`MULTINOMIAL`** — the other variadic integer-combinatorial function in\n this category, and one that shares `GCD`'s group structural evidence record.\n- Readers reach for `GCD` most often to reduce a fraction: `n/GCD(n,d)` over `d/GCD(n,d)`. That\n usage never approaches the documented 2^53 ceiling, which is why the ceiling is so rarely\n noticed.\n\n\n", "how-to-implement": "\n### Numerical notes\n\n\n`GCD` is one of the few functions in the math category with **no floating-point error at all**\nin its intended domain — every intermediate value is an exact integer — and yet it has a\ngenuine numerical hazard, which is the representation boundary rather than rounding.\n\n**The 2^53 wall.** Arguments arrive as binary64. Below 2^53 every integer is exactly\nrepresentable and truncation recovers exactly the integer the user typed. At or above 2^53 the\ndoubles are spaced two or more apart, so the value that arrives is not necessarily the integer\nthat was written, and a gcd computed from it is the gcd of a rounded neighbour. Microsoft's\ndocumented `#NUM!` at 2^53 is precisely a refusal to answer a question that can no longer be\nasked. An implementation that instead converts to a 64-bit integer inherits a *different*\nboundary at 2^63, and above that the conversion saturates or wraps — which is how a function\nwith no rounding error still produces a wrong answer.\n\n**Algorithm choice.** Euclid's algorithm with remainder (Knuth, *TAOCP* volume 2, section 4.5.2)\nruns in O(log min(*a*, *b*)) divisions; the binary GCD of Stein replaces division with shifts\nand subtraction and is usually faster on hardware without a fast integer divide. Both are exact,\nso the choice is performance-only — unusually for this category, there is no accuracy trade-off\nto weigh. The variadic fold benefits from an early exit: once the running gcd reaches 1 no later\nargument can change it, and the remaining arguments need only be validated, not divided.\n\n**What a careful implementation does about the boundary**, in order of decreasing preference:\nguard the documented domain explicitly and return `#NUM!`; or, if a wider domain is wanted,\ncarry the reduction in a type wide enough to hold every finite double's integer part and state\nthe widened domain as a deliberate, documented departure. What it must not do is convert to a\nfixed-width integer and let the conversion decide.\n\n\n", "quirks": "\n### Result and edge cases\n\n\nReturns `Number` — a non-negative integer value.\n\n- **Zero alone** yields zero, consistent with the identity-element reading above.\n- **A logical argument** is accepted and converted; **numeric text** is accepted and then\n truncated. Neither case is covered by Microsoft's page, which speaks only of \"nonnumeric\".\n- **An inline array argument** is consumed as a set of values and reduced to one result, not\n lifted elementwise. This is the aggregate shape, not the scalar-kernel shape.\n- **An empty range** produces `#VALUE!` rather than the zero that the identity element would\n suggest.\n- **A subnormal magnitude** truncates to zero and therefore contributes nothing.\n\n\n\n### Errors\n\n\nAs documented on Microsoft's `GCD` page:\n\n| Error | Documented condition |\n|---|---|\n| `#VALUE!` | \"If any argument is nonnumeric, GCD returns the #VALUE! error value.\" |\n| `#NUM!` | \"If any argument is less than zero, GCD returns the #NUM! error value.\" |\n| `#NUM!` | \"If a parameter to GCD is >=2^53, GCD returns the #NUM! error value.\" |\n\nError values in any argument propagate under the shared coercion discipline.\n\n", "sources": "- Microsoft, \"GCD function\" —\n .\n Retrieved for this page: the syntax, the \"1 to 255 values\" count, the truncation rule, the\n `#VALUE!` nonnumeric remark, the `#NUM!` negative remark, the `#NUM!` at 2^53 remark, and the\n \"One divides any value evenly\" / prime-divisor notes.\n- Handbook, [the value universe](../model/01-value-universe.md) and\n [coercion and lifting](../model/02-coercion-and-lifting.md) — value kinds, to-number outcomes,\n error propagation, and the aggregate-versus-lift distinction.\n- D. E. Knuth, *The Art of Computer Programming*, volume 2, section 4.5.2 — Euclid's algorithm,\n its complexity, and the binary (Stein) variant.\n- `data/presence/FUNC.GCD.json` — the implementing module, its Lean companion, and the\n `BUG-FUNC-028` defect stream that the structural record above resolves.\n\n", "version-differences": "The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.\n\n| Excel 16.0 build | Platform and CPU | Captured inputs |\n|---|---|---:|\n| 20430 | Windows, 64-bit, CPU not recorded | 7,966 |\n\nThe record contains no same-input comparison between these environments. Reading different inputs on different builds does not establish either a version change or cross-build agreement.\n\n[Recorded environments and comparison details](https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.GCD.json).", "what-it-does": "\n`GCD(number1, [number2], ...)` returns the greatest common divisor of its arguments after each\nhas been truncated to an integer.\n\nThe mathematics is the oldest algorithm in the book. For integers *a*, *b* not both zero,\ngcd(*a*, *b*) is the largest positive integer dividing both. The definition extends to any\nnumber of arguments by associativity:\n\n gcd(a, b, c) = gcd(gcd(a, b), c)\n\nso the *n*-argument function is a left fold of the binary one, in any order — gcd is\ncommutative and associative, which is why Excel can accept up to 255 arguments without\ndeclaring an evaluation order.\n\nThe structural facts worth carrying:\n\n- **gcd(*a*, 0) = *a***, and **gcd(0, 0) = 0**. Zero is the identity element: every integer\n divides zero, so zero contributes nothing to the meet. This is not a special case bolted on;\n it is what makes gcd a well-defined operation on the whole of the non-negative integers.\n- **gcd(*a*, 1) = 1.** Microsoft's page states the same fact in its own words — \"One divides any\n value evenly\" — and notes that \"A prime number has only itself and one as even divisors.\"\n- **gcd and lcm are dual.** Under divisibility, the non-negative integers form a lattice in\n which gcd is the meet and lcm the join. For exactly two arguments,\n gcd(*a*, *b*) · lcm(*a*, *b*) = *a* · *b*. **That identity does not extend to three or more\n arguments**, and readers who assume it does are the main source of wrong `LCM` formulas.\n- **Bézout.** gcd(*a*, *b*) is the smallest positive integer expressible as *ax* + *by*. Excel\n exposes no function that returns the Bézout coefficients, so the extended algorithm is not\n reachable from the worksheet.\n\nDomain: non-negative integers after truncation. Range: non-negative integers.\n\n\n\n### Arguments\n\n\n| Argument | Meaning | Required |\n|---|---|---|\n| `number1` | The first value. | Yes |\n| `number2`, … | Further values. | No |\n\nTwo properties of the truncation rule deserve emphasis.\n\n1. **Truncation happens before the arithmetic, not after.** `GCD(2.5, 5)` is the gcd of 2 and\n 5, not a rounding of some fractional gcd. Fractional arguments are silently reinterpreted;\n there is no warning.\n2. **The documented rule says \"truncated\", not \"rounded\".** For non-negative inputs — the only\n ones `GCD` accepts — truncation toward zero and floor agree, so the two readings cannot be\n distinguished on the admissible domain. The distinction that bites elsewhere in the math\n category (see [INT](FUNC.INT.md)) is invisible here.\n\nArgument slots are numeric and subject to ordinary to-number coercion; see\n[coercion and lifting](../model/02-coercion-and-lifting.md). Range arguments are scanned rather\nthan lifted — `GCD` is declared with a custom coercion profile rather than the scalar-lift\nprofile used by single-argument math functions.\n\n\n" }, "signature": { "display": "GCD(number1, [number2], ...)", "parameters": [ { "description": null, "name": "number1", "optional": false, "repeats": false }, { "description": null, "name": "number2", "optional": true, "repeats": true } ], "trailing_repeats": true }, "sources": { "metadata": { "admission": { "label": "implemented", "reason": null }, "arity": { "max": 255, "min": 1 }, "category": "Math and trigonometry functions", "classification": { "arg_preparation_profile": "ValuesOnlyPreAdapter", "coercion_lift_profile": "Custom", "determinism_class": "Deterministic", "error_collapse_profile": "None", "fec_dependency_profile": "None", "host_interaction_class": "None", "kernel_signature_class": "Custom", "rich_value_usage": "RichBlind", "surface_fec_dependency_profile": "RefOnly", "thread_safety_class": "SafePure", "volatility_class": "NonVolatile" }, "descriptions": { "long": null, "short": "Greatest common divisor of whole numbers", "short_source": "dna-calc-own" }, "docs": { "english_description": "Returns the greatest common divisor", "microsoft_url": "https://support.microsoft.com/en-us/office/gcd-function-d5107a51-69e3-461f-8e4c-ddfc21b5073a" }, "entry_kind": "built_in_function", "localized_family_names": null, "localized_names": [ { "locale": "ar-SA", "name": "GCD" }, { "locale": "bg-BG", "name": "GCD" }, { "locale": "cs-CZ", "name": "GCD" }, { "locale": "da-DK", "name": "STØRSTE.FÆLLES.DIVISOR" }, { "locale": "de-DE", "name": "GCD" }, { "locale": "el-GR", "name": "GCD" }, { "locale": "en-AU", "name": "GCD" }, { "locale": "en-GB", "name": "GCD" }, { "locale": "en-US", "name": "GCD" }, { "locale": "es-ES", "name": "M.C.D" }, { "locale": "et-EE", "name": "GCD" }, { "locale": "fi-FI", "name": "SUURIN.YHT.TEKIJÄ" }, { "locale": "fr-FR", "name": "PGCD*" }, { "locale": "he-IL", "name": "GCD" }, { "locale": "hr-HR", "name": "GCD" }, { "locale": "hu-HU", "name": "LKO" }, { "locale": "id-ID", "name": "GCD" }, { "locale": "it-IT", "name": "GCD" }, { "locale": "ja-JP", "name": "GCD" }, { "locale": "ko-KR", "name": "GCD" }, { "locale": "lt-LT", "name": "GCD" }, { "locale": "lv-LV", "name": "GCD" }, { "locale": "nb-NO", "name": "GCD" }, { "locale": "nl-NL", "name": "GGD" }, { "locale": "pl-PL", "name": "NAJW.WSP.DZIEL" }, { "locale": "pt-BR", "name": "GCD" }, { "locale": "pt-PT", "name": "MDC" }, { "locale": "ro-RO", "name": "GCD" }, { "locale": "ru-RU", "name": "НОД" }, { "locale": "sk-SK", "name": "GCD" }, { "locale": "sl-SI", "name": "GCD" }, { "locale": "sr-Latn-RS", "name": "GCD" }, { "locale": "sv-SE", "name": "SGD" }, { "locale": "th-TH", "name": "GCD" }, { "locale": "tr-TR", "name": "OBEB işlevi" }, { "locale": "uk-UA", "name": "GCD" }, { "locale": "vi-VN", "name": "GCD" }, { "locale": "zh-CN", "name": "GCD" }, { "locale": "zh-HK", "name": "GCD" }, { "locale": "zh-TW", "name": "GCD" } ], "signature": { "display": "GCD(number1, [number2], ...)", "parameters": [ { "description": null, "name": "number1", "optional": false, "repeats": false }, { "description": null, "name": "number2", "optional": true, "repeats": true } ], "trailing_repeats": true }, "surface_name": "GCD", "version_marker": null, "xlcall": { "code": "473", "symbol": "xlfGcd" } }, "microsoft": "https://support.microsoft.com/en-us/office/gcd-function-d5107a51-69e3-461f-8e4c-ddfc21b5073a" }, "try_it": { "formula": "=GCD(2.5, 5)" }, "url": "https://handbook.dnacalc.com/functions/GCD.html", "variants": [], "versions": [ { "behaviour": "GCD(9007199254740992, 1) returned 1 (0x3ff0000000000000). GCD(9007199254740994, 0) returned #NUM!. Inputs were supplied through cells using Value2. These are controls from this environment, not a same-input comparison with another build.", "excel_build": "16.0 build 20430", "observations": [ { "call": "GCD(9007199254740992, 1)", "excel": { "bits": "0x3ff0000000000000", "display": "1", "kind": "number" }, "input_route": "cell-value2", "inputs": [ { "name": "number1", "value": { "bits": "0x4340000000000000", "display": "9007199254740992", "kind": "number" } }, { "name": "number2", "value": { "bits": "0x3ff0000000000000", "display": "1", "kind": "number" } } ], "recorded_in": "Excel 16.0 build 20430, Windows 64-bit, CPU not recorded in this capture" }, { "call": "GCD(9007199254740994, 0)", "excel": { "bits": null, "display": "#NUM!", "kind": "error" }, "input_route": "cell-value2", "inputs": [ { "name": "number1", "value": { "bits": "0x4340000000000001", "display": "9007199254740994", "kind": "number" } }, { "name": "number2", "value": { "bits": "0x0000000000000000", "display": "0", "kind": "number" } } ], "recorded_in": "Excel 16.0 build 20430, Windows 64-bit, CPU not recorded in this capture" } ], "platform": "Windows 64-bit, CPU not recorded in this capture", "sources": [ "https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/3345a87/content/notes/FN-GCD-001.json" ] }, { "behaviour": "7,966 captured inputs are indexed for this environment. Capture dates: 2026-09-29 to 2026-09-29. This is a reading count, not a comparison with another build.", "excel_build": "16.0 build 20430", "platform": "Windows, 64-bit, CPU not recorded", "sources": [ "https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.GCD.json" ] } ] }