{ localUrl: '../page/greatest_common_divisor.html', arbitalUrl: 'https://arbital.com/p/greatest_common_divisor', rawJsonUrl: '../raw/5mw.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'greatest_common_divisor', edit: '3', editSummary: 'HCF -> GCD as discussed on slack, US standards', prevEdit: '1', currentEdit: '3', wasPublished: 'true', type: 'wiki', title: 'Greatest common divisor', clickbait: 'The greatest common divisor of two natural numbers is… the largest number which is a divisor of both. The clue is in the name, really.', textLength: '1120', alias: 'greatest_common_divisor', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'EricBruylant', editCreatedAt: '2016-08-04 20:04:44', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-07-28 21:02:31', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '4', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '59', text: '[summary: The greatest common divisor of two [45h natural numbers] is the largest number which divides both of them.]\n\nThere are two ways to define the **greatest common divisor** (also known as **greatest common factor**, or **highest common factor**), both equivalent.\n\nThe first definition is as the name suggests: the GCD of $a$ and $b$ is the largest number which divides both $a$ and $b$.\n\nThe second definition is the more "mathematical", because it generalises to arbitrary [3gq rings] rather than just [55j ordered rings].\nThe GCD of $a$ and $b$ is the number $c$ such that $c \\mid a$, $c \\mid b$, and whenever $d \\mid a$ and $d \\mid b$, we have $d \\mid c$.\n(That is, it is the maximal element of the [-3rb] that consists of the divisors of $a$ and $b$, ordered by division.)\n\n# Examples\n\n[todo: show the two different definitions in action and how they prepare]\n\n# Equivalence of the definitions\n\n[todo: prove this]\n\n# Relation to prime factorisations\n\n[todo: algorithm given access to prime factorisations; explain why this is unhelpful]\n\n# Calculating the GCD efficiently\n\n[todo: link to Euclidean algorithm]', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'PatrickStevens', 'EricBruylant' ], childIds: [ 'bezout_theorem' ], parentIds: [ 'math' ], commentIds: [], questionIds: [], tagIds: [ 'needs_parent_meta_tag', 'stub_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18975', pageId: 'greatest_common_divisor', userId: 'EricBruylant', edit: '0', type: 'newChild', createdAt: '2016-08-20 13:06:06', auxPageId: 'bezout_theorem', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18372', pageId: 'greatest_common_divisor', userId: 'EricBruylant', edit: '0', type: 'newAlias', createdAt: '2016-08-04 20:04:44', auxPageId: '', oldSettingsValue: 'highest_common_factor', newSettingsValue: 'greatest_common_divisor' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18373', pageId: 'greatest_common_divisor', userId: 'EricBruylant', edit: '3', type: 'newEdit', createdAt: '2016-08-04 20:04:44', auxPageId: '', oldSettingsValue: '', newSettingsValue: 'HCF -> GCD as discussed on slack, US standards' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18214', pageId: 'greatest_common_divisor', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-08-03 15:51:51', auxPageId: 'needs_parent_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17697', pageId: 'greatest_common_divisor', userId: 'PatrickStevens', edit: '0', type: 'newTag', createdAt: '2016-07-28 21:02:34', auxPageId: 'stub_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17696', pageId: 'greatest_common_divisor', userId: 'PatrickStevens', edit: '0', type: 'newParent', createdAt: '2016-07-28 21:02:33', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17694', pageId: 'greatest_common_divisor', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-07-28 21:02:31', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'true', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: { improveStub: { likeableId: '3421', likeableType: 'contentRequest', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '59', pageId: 'greatest_common_divisor', requestType: 'improveStub', createdAt: '2016-08-20 13:07:16' } } }