{ "category": "Math and trigonometry functions", "display_name": "LCM", "entry_kind": "built_in_function", "eval_api": { "method": "POST", "url": "https://handbook.dnacalc.com/api/eval" }, "function_id": "FUNC.LCM", "implementation_state": { "answer_bank": { "excel_builds": [ "20430" ], "larger_numeric": 0, "last_bit": 0, "matched": 7967, "other_mismatch": 0, "refused": 0, "rows": 7967, "unread": 0, "worst_ulp": 0 }, "detail": "Recorded: 7,967 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 7,973 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#L138-L141", "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/ported/gcd_lcm_common.rs#L14-L29", "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/ported/lcm_fn.rs#L4-L16" ], "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#L138-L141", "state": "matches", "suite": null, "summary": "Matches all 7,973 recorded answers" } ], "json_url": "https://handbook.dnacalc.com/api/functions/FUNC.LCM.json", "name": "LCM", "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 divisibility primitive underneath both.\n- **`MULTINOMIAL`** — the other variadic exact-integer function in this\n category, sharing `LCM`'s group structural evidence record and its 2^53-scale\n representability problem.\n- Readers most often reach for `LCM` for common denominators and for period alignment (\"both\n events recur together every `LCM(m, n)` steps\"). Both usages sit far below the documented\n ceiling, which is why the ceiling is rarely met in practice and rarely tested.\n\n\n", "how-to-implement": "\n### Numerical notes\n\n\nLike `GCD`, `LCM` computes exactly in its intended domain — there is no rounding error to\nanalyse — and like `GCD` its real hazard is representability. Unlike `GCD`, the hazard is\nreached from *inside* the admissible domain, because the lcm of admissible arguments is not\nitself guaranteed admissible.\n\n**Compute in the order that does not overflow.** The naive binary step\nlcm(*a*, *b*) = *a* · *b* / gcd(*a*, *b*) forms the product first, and that product can exceed\n2^53 — or exceed the machine word — even when the final answer is comfortably small. The\nstandard remedy is to divide first:\n\n lcm(a, b) = (a / gcd(a, b)) * b\n\nThe division is exact because gcd(*a*, *b*) divides *a*, so no precision is lost, and the\nlargest intermediate is the answer itself rather than the product of the inputs. Any\nimplementation that forms `a*b` before dividing has a smaller usable domain than the one it\nadvertises, and its failures appear as silently wrong values rather than as errors when the\narithmetic wraps.\n\n**The fold accumulates.** With many arguments the running lcm grows monotonically, so an\noverflow check belongs *inside* the fold, tested after each step, not once at the end. Checking\nonly the final value cannot detect an intermediate that already wrapped.\n\n**The 2^53 wall.** Above 2^53 consecutive integers are no longer distinct doubles, so an lcm\nreturned as a binary64 above that point is not the integer it claims to be — it is the nearest\nrepresentable neighbour, and it may not even be a multiple of the inputs. Microsoft's documented\n`#NUM!` is a refusal to hand back a value whose defining property has stopped holding. That is\nthe right instinct, and it is worth stating plainly: above the wall the *postcondition* fails,\nnot merely the precision.\n\nKnuth, *The Art of Computer Programming*, volume 2, section 4.5.2 covers the gcd algorithms\nthis function's inner loop rests on; the divide-before-multiply discipline is standard advice in\nevery treatment of exact integer arithmetic.\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 zero as the absorbing element.\n- **A logical argument** is converted and accepted; **numeric text** is converted and then\n truncated. Microsoft's page addresses neither case, speaking only of \"nonnumeric\".\n- **An inline array** is reduced to one result rather than lifted elementwise.\n- **An empty range** produces `#VALUE!`.\n- **A subnormal magnitude** truncates to zero, and zero absorbs — so a single tiny positive\n argument turns the whole result into zero. That is mathematically right and practically\n surprising, and it is the failure mode to watch for when `LCM` is fed computed values rather\n than typed constants.\n\n\n\n### Errors\n\n\nAs documented on Microsoft's `LCM` page:\n\n| Error | Documented condition |\n|---|---|\n| `#VALUE!` | \"If any argument is nonnumeric, LCM returns the #VALUE! error value.\" |\n| `#NUM!` | \"If any argument is less than zero, LCM returns the #NUM! error value.\" |\n| `#NUM!` | \"If LCM(a,b) >=2^53, LCM returns the #NUM! error value.\" |\n\nError values in any argument propagate under the shared coercion discipline.\n\nNote the wording of the third row and compare it with [GCD](FUNC.GCD.md)'s: `GCD`'s ceiling is\ntested on any **parameter**, `LCM`'s on the **result**. The asymmetry is not an editorial\naccident — a gcd is bounded above by its smallest argument and can never overflow inputs that\nwere themselves admissible, whereas an lcm of two modest coprime arguments is their product and\nreaches 2^53 quickly. `LCM` is therefore the member of the pair that must test *after* computing,\nwhich means it must either detect the overflow during the fold or compute in a wider type.\n\n", "sources": "- Microsoft, \"LCM 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!` on the result at 2^53,\n and the common-denominator motivation.\n- Handbook, [coercion and lifting](../model/02-coercion-and-lifting.md) — to-number outcomes,\n the aggregate-versus-lift distinction, error propagation.\n- D. E. Knuth, *The Art of Computer Programming*, volume 2, section 4.5.2 — gcd algorithms and\n exact integer reduction.\n\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 | 7,967 |\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.LCM.json).", "what-it-does": "\n`LCM(number1, [number2], ...)` returns the least common multiple of its arguments after each\nhas been truncated to an integer.\n\nFor positive integers *a*, *b*, lcm(*a*, *b*) is the smallest positive integer that both\ndivide. As with gcd, the *n*-argument function is the associative fold:\n\n lcm(a, b, c) = lcm(lcm(a, b), c)\n\nThe structural facts worth carrying:\n\n- **lcm(*a*, 0) = 0**, and lcm(*a*, 1) = *a*. Zero is the absorbing element of the join, one is\n its identity — exactly the mirror of gcd, where zero is the identity.\n- **The two-argument identity** gcd(*a*, *b*) · lcm(*a*, *b*) = *a* · *b* holds for\n non-negative integers. **It does not generalize to three or more arguments**: the product of\n gcd and lcm of a triple is not the product of the triple. The correct three-argument identity\n runs through the inclusion–exclusion of prime exponents, not through a single product, and\n every attempt to compute a three-way lcm as \"product over gcd\" is wrong. This is the single\n most common error in worksheet code that reimplements `LCM`.\n- **In prime-exponent form**, lcm takes the pointwise maximum of the exponent vectors and gcd\n the pointwise minimum. That is the whole theory: gcd is the meet, lcm the join, of the\n divisibility lattice.\n- Microsoft's page gives the practical motivation: the least common multiple is what you need to\n add fractions with unlike denominators.\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\nTruncation happens before the arithmetic. `LCM(2.9, 3)` is the lcm of 2 and 3. On the\nadmissible non-negative domain, truncation and floor agree, so the negative-number ambiguity\nthat separates truncation from flooring elsewhere in this category cannot arise here.\n\nArgument slots are numeric and subject to ordinary to-number coercion\n([coercion and lifting](../model/02-coercion-and-lifting.md)). Range and array arguments are\nscanned and reduced to a single result; `LCM` is declared with a custom coercion profile, not\nthe scalar-lift profile, so it does not apply elementwise.\n\n\n" }, "signature": { "display": "LCM(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": "Least common multiple of whole numbers", "short_source": "dna-calc-own" }, "docs": { "english_description": "Returns the least common multiple", "microsoft_url": "https://support.microsoft.com/en-us/office/lcm-function-7152b67a-8bb5-4075-ae5c-06ede5563c94" }, "entry_kind": "built_in_function", "localized_family_names": null, "localized_names": [ { "locale": "ar-SA", "name": "LCM" }, { "locale": "bg-BG", "name": "LCM" }, { "locale": "cs-CZ", "name": "LCM" }, { "locale": "da-DK", "name": "MINDSTE.FÆLLES.MULTIPLUM" }, { "locale": "de-DE", "name": "LCM" }, { "locale": "el-GR", "name": "LCM" }, { "locale": "en-AU", "name": "LCM" }, { "locale": "en-GB", "name": "LCM" }, { "locale": "en-US", "name": "LCM" }, { "locale": "es-ES", "name": "M.C.M" }, { "locale": "et-EE", "name": "LCM" }, { "locale": "fi-FI", "name": "PIENIN.YHT.JAETTAVA" }, { "locale": "fr-FR", "name": "PPCM*" }, { "locale": "he-IL", "name": "LCM" }, { "locale": "hr-HR", "name": "LCM" }, { "locale": "hu-HU", "name": "LKT" }, { "locale": "id-ID", "name": "LCM" }, { "locale": "it-IT", "name": "LCM" }, { "locale": "ja-JP", "name": "LCM" }, { "locale": "ko-KR", "name": "LCM" }, { "locale": "lt-LT", "name": "LCM" }, { "locale": "lv-LV", "name": "LCM" }, { "locale": "nb-NO", "name": "LCM" }, { "locale": "nl-NL", "name": "KGV" }, { "locale": "pl-PL", "name": "NAJMN.WSP.WIEL" }, { "locale": "pt-BR", "name": "LCM" }, { "locale": "pt-PT", "name": "MMC" }, { "locale": "ro-RO", "name": "LCM" }, { "locale": "ru-RU", "name": "НОК" }, { "locale": "sk-SK", "name": "LCM" }, { "locale": "sl-SI", "name": "LCM" }, { "locale": "sr-Latn-RS", "name": "LCM" }, { "locale": "sv-SE", "name": "MGM" }, { "locale": "th-TH", "name": "LCM" }, { "locale": "tr-TR", "name": "OKEK işlevi" }, { "locale": "uk-UA", "name": "LCM" }, { "locale": "vi-VN", "name": "LCM" }, { "locale": "zh-CN", "name": "LCM" }, { "locale": "zh-HK", "name": "LCM" }, { "locale": "zh-TW", "name": "LCM" } ], "signature": { "display": "LCM(number1, [number2], ...)", "parameters": [ { "description": null, "name": "number1", "optional": false, "repeats": false }, { "description": null, "name": "number2", "optional": true, "repeats": true } ], "trailing_repeats": true }, "surface_name": "LCM", "version_marker": null, "xlcall": { "code": "475", "symbol": "xlfLcm" } }, "microsoft": "https://support.microsoft.com/en-us/office/lcm-function-7152b67a-8bb5-4075-ae5c-06ede5563c94" }, "try_it": { "formula": "=LCM(2.9, 3)" }, "url": "https://handbook.dnacalc.com/functions/LCM.html", "variants": [], "versions": [ { "behaviour": "LCM(9007199254740992, 2) returned 9007199254740992 (0x4340000000000000). 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": "LCM(9007199254740992, 2)", "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" } ], "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": "7,967 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.LCM.json" ] } ] }