See greatest common divisor on Wiktionary
Download JSON data for greatest common divisor meaning in All languages combined (9.2kB)
{
"antonyms": [
{
"sense": "antonym(s) of \"largest number that divides each of a set of positive integers\"",
"word": "least common multiple"
}
],
"forms": [
{
"form": "greatest common divisors",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "greatest common divisor (plural greatest common divisors)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "English entries with topic categories using raw markup",
"parents": [
"Entries with topic categories using raw markup",
"Entry maintenance"
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"source": "w"
},
{
"kind": "other",
"name": "English terms with non-redundant non-automated sortkeys",
"parents": [
"Terms with non-redundant non-automated sortkeys",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "Mandarin terms with redundant transliterations",
"parents": [
"Terms with redundant transliterations",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Arithmetic",
"orig": "en:Arithmetic",
"parents": [
"Applied sciences",
"Mathematics",
"Sciences",
"Formal sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Number theory",
"orig": "en:Number theory",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"examples": [
{
"text": "The greatest common divisor of 66, 30 and 18 is 6.",
"type": "example"
},
{
"text": "1974, John M. Peterson, Basic Concepts of Elementary Mathematics, Prindle, Weber & Schmidt, page 148,\nEuclid's algorithm is a process for finding the greatest common divisor of any two whole numbers."
},
{
"ref": "2006, W. B. Vasantha Kandasamy, Florentin Smarandache, Neutrosophic Rings, Hexis, page 53",
"text": "Suppose (a#x5F;0,a#x5F;1,#x5C;dots,a#x5F;n)#x5C;in#x5C;Z we say (a#x5F;0,a#x5F;1,#x5C;dots,a#x5F;n) is primitive if the greatest common divisor of a#x5F;0,a#x5F;1,#x5C;dots,a#x5F;n is 1.",
"type": "quotation"
},
{
"ref": "2011, Zhonggang Zeng, “The Numerical Greatest Common Divisor of Univariate Polynomials”, in Leonid Gurvits, Philippe Pébay, J. Maurice Rojas, David Thompson, editors, Randomization, Relaxation, and Complexity in Polynomial Equation Solving, American Mathematical Society, page 187",
"text": "This paper presents a regularization theory for numerical computation of polynomial greatest common divisors and a convergence analysis, along with a detailed description of a blackbox-type algorithm.[…]As one of the fundamental algebraic problems with a long history, finding the greatest common divisor (GCD) of univariate polynomials is an indispensable component of many algebraic computations besides being an important problem in its own right.",
"type": "quotation"
}
],
"glosses": [
"The largest positive integer (respectively polynomial, element of a given ring) that is a divisor of each of a given set of integers (respectively polynomials, elements of a given ring)."
],
"id": "en-greatest_common_divisor-en-noun-8UEMzBKz",
"links": [
[
"arithmetic",
"arithmetic"
],
[
"number theory",
"number theory"
],
[
"integer",
"integer"
],
[
"polynomial",
"polynomial"
],
[
"ring",
"ring"
],
[
"divisor",
"divisor"
]
],
"raw_glosses": [
"(arithmetic, number theory) The largest positive integer (respectively polynomial, element of a given ring) that is a divisor of each of a given set of integers (respectively polynomials, elements of a given ring)."
],
"related": [
{
"word": "maximal common divisor"
}
],
"synonyms": [
{
"sense": "largest number that divides each of a set of positive integers",
"tags": [
"abbreviation"
],
"word": "gcd"
},
{
"sense": "largest number that divides each of a set of positive integers",
"word": "greatest common factor"
},
{
"sense": "largest number that divides each of a set of positive integers",
"word": "highest common factor"
}
],
"topics": [
"arithmetic",
"mathematics",
"number-theory",
"sciences"
],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "zuìdà gōngyīnzǐ",
"sense": "largest positive integer or polynomial",
"word": "最大公因子"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "largest positive integer or polynomial",
"word": "最大公約數"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "zuìdà gōngyuēshù",
"sense": "largest positive integer or polynomial",
"word": "最大公约数"
},
{
"code": "cs",
"lang": "Czech",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "největší společný dělitel"
},
{
"code": "da",
"lang": "Danish",
"sense": "largest positive integer or polynomial",
"word": "største fælles divisor"
},
{
"code": "nl",
"lang": "Dutch",
"sense": "largest positive integer or polynomial",
"word": "grootste gemene deler"
},
{
"code": "et",
"lang": "Estonian",
"sense": "largest positive integer or polynomial",
"word": "suurim ühiskordne"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "largest positive integer or polynomial",
"word": "suurin yhteinen tekijä"
},
{
"code": "fr",
"lang": "French",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "plus grand commun dénominateur"
},
{
"code": "fr",
"lang": "French",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "plus grand commun diviseur"
},
{
"code": "ka",
"lang": "Georgian",
"roman": "udidesi saerto gamq̇opi",
"sense": "largest positive integer or polynomial",
"word": "უდიდესი საერთო გამყოფი"
},
{
"code": "de",
"lang": "German",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "größter gemeinsamer Teiler"
},
{
"code": "el",
"lang": "Greek",
"roman": "mégistos koinós diairétis",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "μέγιστος κοινός διαιρέτης"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "largest positive integer or polynomial",
"word": "legnagyobb közös osztó"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "stærsti samdeilir"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "stærsti sameiginlegi deilir"
},
{
"code": "id",
"lang": "Indonesian",
"sense": "largest positive integer or polynomial",
"word": "faktor persekutuan terbesar"
},
{
"code": "ja",
"english": "integer",
"lang": "Japanese",
"roman": "saidai kōyakusū",
"sense": "largest positive integer or polynomial",
"word": "最大公約数"
},
{
"code": "ja",
"english": "polynomial",
"lang": "Japanese",
"roman": "saidai kōyaku takōshiki",
"sense": "largest positive integer or polynomial",
"word": "最大公約多項式"
},
{
"code": "ko",
"lang": "Korean",
"roman": "choedaegong'yaksu",
"sense": "largest positive integer or polynomial",
"word": "최대공약수"
},
{
"code": "ms",
"lang": "Malay",
"sense": "largest positive integer or polynomial",
"word": "faktor sepunya terbesar"
},
{
"code": "ml",
"lang": "Malayalam",
"roman": "uttama sādhāraṇa ghaṭakaṁ",
"sense": "largest positive integer or polynomial",
"word": "ഉത്തമ സാധാരണ ഘടകം"
},
{
"code": "no",
"lang": "Norwegian",
"sense": "largest positive integer or polynomial",
"word": "største felles faktor"
},
{
"code": "no",
"lang": "Norwegian",
"sense": "largest positive integer or polynomial",
"word": "største felles nevner"
},
{
"code": "pl",
"lang": "Polish",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "największy wspólny dzielnik"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "máximo divisor comum"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "largest positive integer or polynomial",
"word": "cel mai mare divizor comun"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "largest positive integer or polynomial",
"word": "c.m.m.d.c"
},
{
"code": "ru",
"lang": "Russian",
"roman": "naibólʹšij óbščij delítelʹ",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "наибо́льший о́бщий дели́тель"
},
{
"code": "es",
"lang": "Spanish",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "máximo común divisor"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "largest positive integer or polynomial",
"word": "största gemensamma delare"
},
{
"code": "tr",
"lang": "Turkish",
"sense": "largest positive integer or polynomial",
"word": "en büyük ortak bölen"
},
{
"code": "tr",
"lang": "Turkish",
"sense": "largest positive integer or polynomial",
"word": "ortak bölenlerin en büyüğü"
},
{
"code": "uk",
"lang": "Ukrainian",
"roman": "najbílʹšyj spílʹnyj dilʹnýk",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "найбі́льший спі́льний дільни́к"
},
{
"code": "vi",
"lang": "Vietnamese",
"sense": "largest positive integer or polynomial",
"word": "ước chung lớn nhất"
},
{
"code": "vi",
"lang": "Vietnamese",
"sense": "largest positive integer or polynomial",
"word": "ước số chung lớn nhất"
}
],
"wikipedia": [
"greatest common divisor"
]
}
],
"word": "greatest common divisor"
}
{
"antonyms": [
{
"sense": "antonym(s) of \"largest number that divides each of a set of positive integers\"",
"word": "least common multiple"
}
],
"forms": [
{
"form": "greatest common divisors",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "greatest common divisor (plural greatest common divisors)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "maximal common divisor"
}
],
"senses": [
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English entries with topic categories using raw markup",
"English lemmas",
"English multiword terms",
"English nouns",
"English terms with non-redundant non-automated sortkeys",
"English terms with quotations",
"English terms with usage examples",
"Mandarin terms with redundant transliterations",
"en:Arithmetic",
"en:Number theory"
],
"examples": [
{
"text": "The greatest common divisor of 66, 30 and 18 is 6.",
"type": "example"
},
{
"text": "1974, John M. Peterson, Basic Concepts of Elementary Mathematics, Prindle, Weber & Schmidt, page 148,\nEuclid's algorithm is a process for finding the greatest common divisor of any two whole numbers."
},
{
"ref": "2006, W. B. Vasantha Kandasamy, Florentin Smarandache, Neutrosophic Rings, Hexis, page 53",
"text": "Suppose (a#x5F;0,a#x5F;1,#x5C;dots,a#x5F;n)#x5C;in#x5C;Z we say (a#x5F;0,a#x5F;1,#x5C;dots,a#x5F;n) is primitive if the greatest common divisor of a#x5F;0,a#x5F;1,#x5C;dots,a#x5F;n is 1.",
"type": "quotation"
},
{
"ref": "2011, Zhonggang Zeng, “The Numerical Greatest Common Divisor of Univariate Polynomials”, in Leonid Gurvits, Philippe Pébay, J. Maurice Rojas, David Thompson, editors, Randomization, Relaxation, and Complexity in Polynomial Equation Solving, American Mathematical Society, page 187",
"text": "This paper presents a regularization theory for numerical computation of polynomial greatest common divisors and a convergence analysis, along with a detailed description of a blackbox-type algorithm.[…]As one of the fundamental algebraic problems with a long history, finding the greatest common divisor (GCD) of univariate polynomials is an indispensable component of many algebraic computations besides being an important problem in its own right.",
"type": "quotation"
}
],
"glosses": [
"The largest positive integer (respectively polynomial, element of a given ring) that is a divisor of each of a given set of integers (respectively polynomials, elements of a given ring)."
],
"links": [
[
"arithmetic",
"arithmetic"
],
[
"number theory",
"number theory"
],
[
"integer",
"integer"
],
[
"polynomial",
"polynomial"
],
[
"ring",
"ring"
],
[
"divisor",
"divisor"
]
],
"raw_glosses": [
"(arithmetic, number theory) The largest positive integer (respectively polynomial, element of a given ring) that is a divisor of each of a given set of integers (respectively polynomials, elements of a given ring)."
],
"topics": [
"arithmetic",
"mathematics",
"number-theory",
"sciences"
],
"wikipedia": [
"greatest common divisor"
]
}
],
"synonyms": [
{
"sense": "largest number that divides each of a set of positive integers",
"tags": [
"abbreviation"
],
"word": "gcd"
},
{
"sense": "largest number that divides each of a set of positive integers",
"word": "greatest common factor"
},
{
"sense": "largest number that divides each of a set of positive integers",
"word": "highest common factor"
}
],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "zuìdà gōngyīnzǐ",
"sense": "largest positive integer or polynomial",
"word": "最大公因子"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "largest positive integer or polynomial",
"word": "最大公約數"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "zuìdà gōngyuēshù",
"sense": "largest positive integer or polynomial",
"word": "最大公约数"
},
{
"code": "cs",
"lang": "Czech",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "největší společný dělitel"
},
{
"code": "da",
"lang": "Danish",
"sense": "largest positive integer or polynomial",
"word": "største fælles divisor"
},
{
"code": "nl",
"lang": "Dutch",
"sense": "largest positive integer or polynomial",
"word": "grootste gemene deler"
},
{
"code": "et",
"lang": "Estonian",
"sense": "largest positive integer or polynomial",
"word": "suurim ühiskordne"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "largest positive integer or polynomial",
"word": "suurin yhteinen tekijä"
},
{
"code": "fr",
"lang": "French",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "plus grand commun dénominateur"
},
{
"code": "fr",
"lang": "French",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "plus grand commun diviseur"
},
{
"code": "ka",
"lang": "Georgian",
"roman": "udidesi saerto gamq̇opi",
"sense": "largest positive integer or polynomial",
"word": "უდიდესი საერთო გამყოფი"
},
{
"code": "de",
"lang": "German",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "größter gemeinsamer Teiler"
},
{
"code": "el",
"lang": "Greek",
"roman": "mégistos koinós diairétis",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "μέγιστος κοινός διαιρέτης"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "largest positive integer or polynomial",
"word": "legnagyobb közös osztó"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "stærsti samdeilir"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "stærsti sameiginlegi deilir"
},
{
"code": "id",
"lang": "Indonesian",
"sense": "largest positive integer or polynomial",
"word": "faktor persekutuan terbesar"
},
{
"code": "ja",
"english": "integer",
"lang": "Japanese",
"roman": "saidai kōyakusū",
"sense": "largest positive integer or polynomial",
"word": "最大公約数"
},
{
"code": "ja",
"english": "polynomial",
"lang": "Japanese",
"roman": "saidai kōyaku takōshiki",
"sense": "largest positive integer or polynomial",
"word": "最大公約多項式"
},
{
"code": "ko",
"lang": "Korean",
"roman": "choedaegong'yaksu",
"sense": "largest positive integer or polynomial",
"word": "최대공약수"
},
{
"code": "ms",
"lang": "Malay",
"sense": "largest positive integer or polynomial",
"word": "faktor sepunya terbesar"
},
{
"code": "ml",
"lang": "Malayalam",
"roman": "uttama sādhāraṇa ghaṭakaṁ",
"sense": "largest positive integer or polynomial",
"word": "ഉത്തമ സാധാരണ ഘടകം"
},
{
"code": "no",
"lang": "Norwegian",
"sense": "largest positive integer or polynomial",
"word": "største felles faktor"
},
{
"code": "no",
"lang": "Norwegian",
"sense": "largest positive integer or polynomial",
"word": "største felles nevner"
},
{
"code": "pl",
"lang": "Polish",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "największy wspólny dzielnik"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "máximo divisor comum"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "largest positive integer or polynomial",
"word": "cel mai mare divizor comun"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "largest positive integer or polynomial",
"word": "c.m.m.d.c"
},
{
"code": "ru",
"lang": "Russian",
"roman": "naibólʹšij óbščij delítelʹ",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "наибо́льший о́бщий дели́тель"
},
{
"code": "es",
"lang": "Spanish",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "máximo común divisor"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "largest positive integer or polynomial",
"word": "största gemensamma delare"
},
{
"code": "tr",
"lang": "Turkish",
"sense": "largest positive integer or polynomial",
"word": "en büyük ortak bölen"
},
{
"code": "tr",
"lang": "Turkish",
"sense": "largest positive integer or polynomial",
"word": "ortak bölenlerin en büyüğü"
},
{
"code": "uk",
"lang": "Ukrainian",
"roman": "najbílʹšyj spílʹnyj dilʹnýk",
"sense": "largest positive integer or polynomial",
"tags": [
"masculine"
],
"word": "найбі́льший спі́льний дільни́к"
},
{
"code": "vi",
"lang": "Vietnamese",
"sense": "largest positive integer or polynomial",
"word": "ước chung lớn nhất"
},
{
"code": "vi",
"lang": "Vietnamese",
"sense": "largest positive integer or polynomial",
"word": "ước số chung lớn nhất"
}
],
"word": "greatest common divisor"
}
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-04-30 from the enwiktionary dump dated 2024-04-21 using wiktextract (210104c and c9440ce). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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