{"trustable":true,"sections":[{"title":"","value":{"format":"HTML","content":" Andrew has just made a breakthrough in group theory: he realized that he can classify all finite Abelian groups (not much of a breakthrough, indeed). Given \u003ci\u003en\u003c/i\u003e, how many Abelian groups with \u003ci\u003en\u003c/i\u003e elements exist up to isomorphism? To help you solve this problem we provide some definitions and theorems from basic algebra (most are cited from Wikipedia). An abelian group is a set, \u003ci\u003eA\u003c/i\u003e, together with an operation \u0027·\u0027 that combines any two elements \u003ci\u003ea\u003c/i\u003e and \u003ci\u003eb\u003c/i\u003e to form another element denoted \u003ci\u003ea\u003c/i\u003e · \u003ci\u003eb\u003c/i\u003e. The symbol \u0027·\u0027 is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, (\u003ci\u003eA\u003c/i\u003e, ·), must satisfy five requirements known as the abelian group axioms: \u003cul\u003e \u003cli\u003e\u003ci\u003eClosure\u003c/i\u003e: for all \u003ci\u003ea\u003c/i\u003e, \u003ci\u003eb\u003c/i\u003e in \u003ci\u003eA\u003c/i\u003e, the result of the operation \u003ci\u003ea\u003c/i\u003e · \u003ci\u003eb\u003c/i\u003e is also in \u003ci\u003eA\u003c/i\u003e. \u003c/li\u003e\u003cli\u003e\u003ci\u003eAssociativity\u003c/i\u003e: for all \u003ci\u003ea\u003c/i\u003e, \u003ci\u003eb\u003c/i\u003e and \u003ci\u003ec\u003c/i\u003e in \u003ci\u003eA\u003c/i\u003e, the equation (\u003ci\u003ea\u003c/i\u003e · \u003ci\u003eb\u003c/i\u003e) · \u003ci\u003ec\u003c/i\u003e \u003d \u003ci\u003ea\u003c/i\u003e · (\u003ci\u003eb\u003c/i\u003e · \u003ci\u003ec\u003c/i\u003e) holds. \u003c/li\u003e\u003cli\u003e\u003ci\u003eIdentity element\u003c/i\u003e: there exists an element \u003ci\u003ee\u003c/i\u003e in \u003ci\u003eA\u003c/i\u003e, such that for all elements \u003ci\u003ea\u003c/i\u003e in \u003ci\u003eA\u003c/i\u003e, the equation \u003ci\u003ee\u003c/i\u003e · \u003ci\u003ea\u003c/i\u003e \u003d \u003ci\u003ea\u003c/i\u003e · \u003ci\u003ee\u003c/i\u003e \u003d \u003ci\u003ea\u003c/i\u003e holds. \u003c/li\u003e\u003cli\u003e\u003ci\u003eInverse element\u003c/i\u003e: for each \u003ci\u003ea\u003c/i\u003e in \u003ci\u003eA\u003c/i\u003e, there exists an element \u003ci\u003eb\u003c/i\u003e in \u003ci\u003eA\u003c/i\u003e such that \u003ci\u003ea\u003c/i\u003e · \u003ci\u003eb\u003c/i\u003e \u003d \u003ci\u003eb\u003c/i\u003e · \u003ci\u003ea\u003c/i\u003e \u003d \u003ci\u003ee\u003c/i\u003e, where \u003ci\u003ee\u003c/i\u003e is the identity element. \u003c/li\u003e\u003cli\u003e\u003ci\u003eCommutativity\u003c/i\u003e: for all \u003ci\u003ea\u003c/i\u003e, \u003ci\u003eb\u003c/i\u003e in \u003ci\u003eA\u003c/i\u003e, \u003ci\u003ea\u003c/i\u003e · \u003ci\u003eb\u003c/i\u003e \u003d \u003ci\u003eb\u003c/i\u003e · \u003ci\u003ea\u003c/i\u003e. \u003c/li\u003e\u003c/ul\u003e An example of an abelian group is a \u003ci\u003ecyclic group\u003c/i\u003e of order \u003ci\u003en\u003c/i\u003e: the set is integers between 0 and \u003ci\u003en\u003c/i\u003e-1, and the operation is sum modulo \u003ci\u003en\u003c/i\u003e. Given two abelian groups \u003ci\u003eG\u003c/i\u003e and \u003ci\u003eH\u003c/i\u003e, their \u003ci\u003edirect sum\u003c/i\u003e is a group where each element is a pair (\u003ci\u003eg\u003c/i\u003e, \u003ci\u003eh\u003c/i\u003e) with \u003ci\u003eg\u003c/i\u003e from \u003ci\u003eG\u003c/i\u003e and \u003ci\u003eh\u003c/i\u003e from \u003ci\u003eH\u003c/i\u003e, and operations are performed on each element of the pair independently. Two groups \u003ci\u003eG\u003c/i\u003e and \u003ci\u003eH\u003c/i\u003e are \u003ci\u003eisomorphic\u003c/i\u003e when there exists a one-to-one mapping \u003ci\u003ef\u003c/i\u003e from elements of \u003ci\u003eG\u003c/i\u003e to elements of \u003ci\u003eH\u003c/i\u003e such that \u003ci\u003ef\u003c/i\u003e(\u003ci\u003ea\u003c/i\u003e) · \u003ci\u003ef\u003c/i\u003e(\u003ci\u003eb\u003c/i\u003e) \u003d \u003ci\u003ef\u003c/i\u003e(\u003ci\u003ea\u003c/i\u003e · \u003ci\u003eb\u003c/i\u003e) for all \u003ci\u003ea\u003c/i\u003e and \u003ci\u003eb\u003c/i\u003e. The \u003ci\u003efundamental theorem of finite abelian groups\u003c/i\u003e states that every finite abelian group is isomorphic to a direct sum of several cyclic groups. The \u003ci\u003eChinese remainder theorem\u003c/i\u003e states that when \u003ci\u003em\u003c/i\u003e and \u003ci\u003en\u003c/i\u003e are coprime, a cyclic group of order \u003ci\u003emn\u003c/i\u003e is isomorphic to the direct sum of the cyclic group of order \u003ci\u003em\u003c/i\u003e and the cyclic group of order \u003ci\u003en\u003c/i\u003e. \u003cbr\u003e\u003cdiv align\u003d\"left\" style\u003d\"margin-top: 1.0em;\"\u003e\u003cb\u003eInput\u003c/b\u003e\u003c/div\u003eFirst and only line of the input file contains an integer \u003ci\u003en\u003c/i\u003e, 1 ≤ \u003ci\u003en\u003c/i\u003e ≤ 10\u003csup\u003e12\u003c/sup\u003e. \u003cbr\u003e\u003cdiv align\u003d\"left\" style\u003d\"margin-top: 1.0em;\"\u003e\u003cb\u003eOutput\u003c/b\u003e\u003c/div\u003eIn the only line of the output file write the number of abelian groups with \u003ci\u003en\u003c/i\u003e elements. \u003cbr\u003e"}},{"title":"Sample 1","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e5\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e1\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}},{"title":"Sample 2","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e4\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e2\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}},{"title":"Sample 3","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e12\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e2\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}}]}