{"text_format":"1","workbook":"xlsx","n_sheets":6,"sheets":["Euclides","Algoritmo","AX=B","Anillo Zn","Teorema chino","Homomorfismo"],"not_rendered":{"charts":0,"images":0,"macros":1,"external_links":0,"defined_names":4,"comments":0,"pivot_tables":0},"unopened":[],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"Euclides","index":0,"n_rows":36,"n_cols":5,"rows_kept":36,"cut":false,"hidden":false,"merged":[[2,1,5,2],[2,4,5,12]]}
["A. Roldán 2012"]
[]
[null,"No se debe alterar el orden de las hojas, porque no funcionarían bien los botones",null,null,"La herencia de Euclides"]
[]
[]
[]
[]
[]
[null,null,null,"En esta hoja de cálculo se incluyen varias técnicas basadas en el algoritmo de Euclides"]
[null,null,null,"Se presenta el algoritmo alojado en un conjunto de celdas y también como una colección de varias funciones"]
[null,null,null,"Los procedimientos basados en este algoritmos abarcarán desde la resolución de ecuaciones lineales en congruencias hasta la ecuación de Pell"]
[]
[null,null,null,"Distribución por hojas"]
[]
[null,null,"Algoritmo",null,"Desarrolla el algoritmo extendido del M.C.D. Y muestra las funciones internas a que da lugar"]
[null,null,null,null,"Son las siguientes: "]
[null,null,null,null,"NCOCIENTES representa el número de cocientes del algoritmo, MCD al máximo común divisor"]
[null,null,null,null,"COC es la matriz de todos los cocientes ordenados."]
[null,null,null,null,"NUM y DEN son los numeradores y denominadores respectivamente de las convergentes de la fracción A/B"]
[]
[]
[null,null,"Ecuación lineal",null,"Si A es primo con m, existe una sola solución x =B*A-1 (mod m), por ser A inversible"]
[null,null,"AX=B (mod m",null,"Si MCD(A,m)=d, con d mayor que 1, para que exista solución ha de ser B múltiplo de d."]
[null,null,null,null,"En ese caso se simplifican los tres números A,B y m con lo que se pasa al primer caso. "]
[null,null,null,null,"Se puede encontrar una primera solución x 0= B*A-1 (mod m) y existirán en total d soluciones, que vienen dadas por la fórmula xr = x 0 +r*m/d"]
[]
[null,null,"Anillo Zm",null,"El algoritmo extendido de Euclides permite el cálculo del inverso de un elemento en el anillo Zm"]
[null,null,null,null,"En primer lugar se determina si es inversible y después se le encuentra el inverso"]
[null,null,null,null,"Como dato se toma el módulo y a partir de él de desarrollan los inversos y la tabla de multiplicar de su grupo."]
[]
[null,null,"Teorema chino",null,"Resuelve el teorema Chino de los restos para varias ecuaciones (entre 2 y 6)"]
[null,null,null,null,"Basta dar los segundos miembros de las ecuaciones y los módulos, y la hoja encuentra todas"]
[null,null,null,null,"las soluciones si existen,"]
[]
[null,null,"Homomorfismo",null,"Presenta en forma de tabla el homomosfirsmo existente entre Zn*Zm y Znm "]
[null,null,null,null,"cuando los módulos son primos entre sí."]

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[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"(-114)* 24075+( 353)* 7775= 25"]
[null,null,null,null,null,"Algoritmo de Euclides"]
[]
[]
[]
[]
[null,null,null,null,"Primer número",null,2910,null,null,"Segundo número",null,8720]
[null,null,null,null,16]
[]
[]
[null,null,null,null,null,null,"Técnica manual, con el uso de las prestaciones de una hoja de cálculo"]
[]
[null,null,{"$f":"IF(B14*C14<>0,INT(B14/C14),0)","v":0},2,1,290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
[null,{"$f":"G7","v":2910},{"$f":"L7","v":8720},{"$f":"IF(B15<>0,B15,0)","v":2910},2900,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
[null,{"$f":"IF(B14*C14<>0,MOD(B14,C14),0)","v":2910},2900,10,0,0,0,0,0,0,0,0,0,0,0,0,{"$f":"IF(Q14*Euclides!S25<>0,MOD(Q14,Euclides!S25),0)","v":0},{"$f":"IF(R14*Euclides!T25<>0,MOD(R14,Euclides!T25),0)","v":0},{"$f":"IF(S14*Euclides!U25<>0,MOD(S14,Euclides!U25),0)","v":0},{"$f":"IF(T14*Euclides!V25<>0,MOD(T14,Euclides!V25),0)","v":0},{"$f":"IF(U14*Euclides!W25<>0,MOD(U14,Euclides!W25),0)","v":0},{"$f":"IF(V14*Euclides!X25<>0,MOD(V14,Euclides!X25),0)","v":0},{"$f":"IF(W14*Euclides!Y25<>0,MOD(W14,Euclides!Y25),0)","v":0},{"$f":"IF(X14*Euclides!Z25<>0,MOD(X14,Euclides!Z25),0)","v":0}]
[]
[null,null,null,null,"Número de cocientes",null,null,{"$f":"COUNTIF(C14:X14,\">0\")","v":4},null,"Máximo común divisor",null,null,{"$f":"INDEX(C14:X14,1,H17)","v":10}]
[]
[null,null,null,null,null,null,"Técnica de funciones definidas. Pulsa el botón para obtenerlas"]
[]
[]
[]
[null,"Cocientes",null,null,"Número de cocientes",null,null,4,null,"M.C.D.",10,null,"Convergentes (o reducidas)"]
[null,null,null,null,"Identidad de Bezout"]
[null,"COC( 1)",0,null,null,null,null,null,null,null,null,"NUM( 1)",0,"DEN( 1)",1]
[null,"COC( 2)",2,null,null,null,null,null,null,null,null,"NUM( 2)",1,"DEN( 2)",0]
[null,"COC( 3)",1,null,"( 3)* 2910+(-1)* 8720= 10",null,null,null,null,null,null,"NUM( 3)",0,"DEN( 3)",1]
[null,"COC( 4)",290,null,null,null,null,null,null,null,null,"NUM( 4)",1,"DEN( 4)",2]
[null,null,null,null,null,null,null,null,null,null,null,"NUM( 5)",1,"DEN( 5)",3]
[null,null,null,null,null,null,null,null,null,null,null,"NUM( 6)",291,"DEN( 6)",872]

{"sheet":"AX=B","index":2,"n_rows":23,"n_cols":10,"rows_kept":23,"cut":false,"hidden":false,"merged":[[2,3,3,7],[7,8,7,10],[8,8,8,10]]}
[]
[]
[null,null,null,"Ecuación lineal Ax=B(m"]
[]
[]
[null,null,null,"Datos",null,null,"Reducidos"]
[]
[null,null,null,"A",22,null,{"$f":"IF($J$10=2,E8/GCD(E$8,E$10),E8)","v":22},null,"Tipo de ecuación"]
[null,null,null,"B",15,null,{"$f":"IF($J$10=2,E9/GCD(E$8,E$10),E9)","v":15},null,{"$f":"IF(J10=1,\"Tiene una sola solución\",IF(J10=2,\"Tiene varias soluciones \",\"No tiene solución\"))","v":"Tiene una sola solución"}]
[null,null,null,"m",17,null,{"$f":"IF($J$10=2,E10/GCD(E$8,E$10),E10)","v":17}," ",{"$f":"GCD(A,m)","v":1},{"$f":"IF(mc=1,1,IF(B/mc=INT(B/mc),2,0))","v":1}]
[]
[]
[]
[]
[null,null,null,"Reducidas"]
[null,null,null,null,null,null,null,"Producto cruzado",null,1]
[null,null,null,"NUM( 1)",0,"DEN( 1)",1,"Penúltimo DEN",null,7]
[null,null,null,"NUM( 2)",1,"DEN( 2)",0]
[null,null,null,"NUM( 3)",1,"DEN( 3)",1,"INV(A) mod m",null,7]
[null,null,null,"NUM( 4)",4,"DEN( 4)",3]
[null,null,null,"NUM( 5)",9,"DEN( 5)",7,"INV(A)*B",null,3]
[null,null,null,"NUM( 6)",22,"DEN( 6)",17]
[null,null,null,null,null,null,null,"Soluciones",null,3]

{"sheet":"Anillo Zn","index":3,"n_rows":32,"n_cols":21,"rows_kept":32,"cut":false,"hidden":false,"merged":[[2,3,3,7]]}
[]
[]
[null,null,null,"Anillo Zn"]
[]
[]
[null,null,null,"Escribe el valor de N",null,21]
[]
[]
[]
[null,null,null,null,null,null,null,null,null,null,null,"Tabla del grupo multiplicativo Z*m"]
[null,null,"X","MCD(X,N)","INVERSO"," ORDEN","Indicatriz de Euler",null,12]
[]
[null,null,1,1,1,1]
[null,null,2,1,11,6,null,null,null,1,2,4,5,8,10,11,13,16,17,19,20]
[null,null,3,3," ",null,null,null,1,1,2,4,5,8,10,11,13,16,17,19,20]
[null,null,4,1,16,3,null,null,2,2,4,8,10,16,20,1,5,11,13,17,19]
[null,null,5,1,17,6,null,null,4,4,8,16,20,11,19,2,10,1,5,13,17]
[null,null,6,3," ",null,null,null,5,5,10,20,4,19,8,13,2,17,1,11,16]
[null,null,7,7," ",null,null,null,8,8,16,11,19,1,17,4,20,2,10,5,13]
[null,null,8,1,8,2,null,null,10,10,20,19,8,17,16,5,4,13,2,1,11]
[null,null,9,3," ",null,null,null,11,11,1,2,13,4,5,16,17,8,19,20,10]
[null,null,10,1,19,6,null,null,13,13,5,10,2,20,4,17,1,19,11,16,8]
[null,null,11,1,2,6,null,null,16,16,11,1,17,2,13,8,19,4,20,10,5]
[null,null,12,3," ",null,null,null,17,17,13,5,1,10,2,19,11,20,16,8,4]
[null,null,13,1,13,2,null,null,19,19,17,13,11,5,1,20,16,10,8,4,2]
[null,null,14,7," ",null,null,null,20,20,19,17,16,13,11,10,8,5,4,2,1]
[null,null,15,3," "]
[null,null,16,1,4,3]
[null,null,17,1,5,6]
[null,null,18,3," "]
[null,null,19,1,10,6]
[null,null,20,1,20,2]

{"sheet":"Teorema chino","index":4,"n_rows":46,"n_cols":13,"rows_kept":46,"cut":false,"hidden":false,"merged":[[2,3,3,7],[2,9,3,10],[24,4,25,6],[29,5,30,7]]}
[]
[]
[null,null,null,"Sistema de congruencias",null,null,null,null,null,"Teorema chino de los restos"]
[]
[]
[null,null,null,"X=B1 (m1 ","X=B2 (m2 ","X=B3 (m3 ","...","X=Bk (mk "]
[]
[null,"Los módulos han de ser primos entre sí dos a dos. La resolución de abajo no es válida si no se cumple esta condición."]
[null,"El valor de X es el mismo en todas las congruencias"]
[]
[null,null,null,"Número de ecuaciones",null,null,2,null,"(No más de seis ni menos de 2)"]
[]
[]
[null,"Valores de B",null,null,null,"Valores de los módulos"]
[null,"(No más de seis y en filas consecutivas)"]
[]
[null,"B1",8,null,null,"M1",13]
[null,"B2",5,null,null,"M2",11]
[null,"B3",0,null,null,"M3",0]
[null,"B4",0,null,null,"M4",0]
[null,"B5",0,null,null,"M5",0]
[null,"B6",0,null,null,"M6",0]
[]
[]
[]
[]
[]
[null,"Llamamos H al producto de los módulos y Hi al cociente de H entre el módulo m i",null,null,null,null,null,null,null,null,null,null," "]
[]
[null,"Valores de Hi",null,"H=",143,"Rsoluciones de las ecuaciones lineales H i * X = 1(m i",null,null,null,"Producto H i * X * B i"]
[]
[null,"H1",11,null,null,6,null,null,null,528]
[null,"H2",13,null,null,6,null,null,null,390]
[null,"H3",0,null,null,0,null,null,null,0]
[null,"H4",0,null,null,0,null,null,null,0]
[null,"H5",0,null,null,0,null,null,null,0]
[null,"H6",0,null,null,0,null,null,null,0]
[]
[null,null,null,null,null,null,null,"Solución primera",null,{"$f":"MOD(SUM(J32:J37),E30)","v":60}]
[]
[null,"Comprobación",null,null,null,null,null,"Otras soluciones",null,{"$f":"J39+E30","v":203}]
[null,null,null,null,null,null,null,null,null,{"$f":"J41+$E$30","v":346}]
[null,"Podemos comprobar si",null,{"$f":"J39","v":60}," es solución",null,null,null,null,{"$f":"J42+$E$30","v":489}]
[null,null,null,null,null,null,null,null,null,{"$f":"J43+$E$30","v":632}]
[null,"Módulos",{"$f":"G17","v":13},{"$f":"G18","v":11},{"$f":"G19","v":0},{"$f":"G20","v":0},{"$f":"G21","v":0},{"$f":"G22","v":0}]
[null,"Restos",{"$f":"IF(C45<>0,MOD($J$39,C45),\" \")","v":8},5," "," "," "," "]

{"sheet":"Homomorfismo","index":5,"n_rows":27,"n_cols":9,"rows_kept":27,"cut":false,"hidden":false,"merged":[[2,3,3,7]]}
[]
[]
[null,null,null,"Homomorfismo de anillos Zm*Zn y Zmn"]
[]
[]
[null,null,"Escribe los dos módulos m y n primos entre sí",null,null,null,null,"m=",5]
[null,null,null,null,null,null,null,"n=",16]
[]
[]
[]
[null,null,null,0,1,2,3,4]
[null,null,0,0,16,32,48,64]
[null,null,1,65,1,17,33,49]
[null,null,2,50,66,2,18,34]
[null,null,3,35,51,67,3,19]
[null,null,4,20,36,52,68,4]
[null,null,5,5,21,37,53,69]
[null,null,6,70,6,22,38,54]
[null,null,7,55,71,7,23,39]
[null,null,8,40,56,72,8,24]
[null,null,9,25,41,57,73,9]
[null,null,10,10,26,42,58,74]
[null,null,11,75,11,27,43,59]
[null,null,12,60,76,12,28,44]
[null,null,13,45,61,77,13,29]
[null,null,14,30,46,62,78,14]
[null,null,15,15,31,47,63,79]