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Multiplos y divisores

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Presentación del tema 4: Múltiplos y divisores.

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Multiplos y divisores

  1. 2. INTRODUCCIÓN A LOS MÚLTIPLOS Laura está haciendo flexiones. Cada 5 flexiones, para un momento para descansar y apunta en una tabla cuántas flexiones lleva hechas hasta el momento. Los números 5, 10,15,20,25 y 30 son múltiplos de 5. Los múltiplos de 5 se pueden calcular de dos formas: <ul><li>Contando de 5 en 5 a partir de 0.
  2. 3. Multiplicando por 5 los números naturales 0,1,2,3,4,5.. . </li></ul>
  3. 4. MÚLTIPLOS DE UN NÚMERO Definición : Un número es múltiplo de otro cuando lo contiene un número exacto de veces, es decir, cuando la división del primero entre el segundo es exacta. 10 es múltiplo de 2 ya que 10 : 2 = 5 y r = 0 Para indicar abreviadamente que un numero es múltiplo de otro Escribiremos: 10 = 2 se lee 10 es múltiplo de 2. Un número tiene infinitos múltiplos. Se obtienen multiplicando sucesivamente el número por los números naturales (0, 1, 2, 3…)
  4. 5. VAMOS A PRACTICAR
  5. 6. MÍNIMO COMÚN MULTIPLO Dados dos números 9 y 12: = {0, 9, 18, 27, 36 , 45, 54, 63, 72 , 81, 90, 99, 108 . .. } = {0, 12, 24, 36 , 48, 60, 72 , 84, 96, 108 ... } Los múltiplos comunes de 9 y 12 seran: /36, 72, 108…/, porque son los múltiplos comunes que coinciden en ambos números exceptuando el cero. Al menor de los múltiplos comunes, 36, lo denominaremos mínimo común múltiplo de 9 y 12. Y lo escribiremos: m.c.m. (9, 12) = 36
  6. 7. Vamos a practicar http://www.gobiernodecanarias.org/educacion/9/Usr/eltanque/todo_mate/multiplosydivisores/mcm/mcm_p.html
  7. 8. Otra forma de calcular el m.c.m.
  8. 9. INTRODUCCIÓN A LOS DVISORES Eduardo va a pegar 8 fotos en su álbum. Quiere poner en cada hoja el mismo número de fotos y que no le sobre ninguna foto.¿Cuántas fotos puede poner en cada hoja? Observa cómo puede repartir Eduardo las fotos en partes iguales en las hojas del álbum.
  9. 10. Eduardo puede poner en cada hoja 1,2,4, u 8 fotos. <ul><li>Los números 1,2,4,y 8 son divisores de 8, porque al dividir 8 entre cada uno de ellos la división es exacta.
  10. 11. Los números 3, 5,6 y 7 no son divisores de 8, porque la dividir 8 entre cada uno de ellos la división es entera. </li></ul>
  11. 12. DIVISORES DE UN NÚMERO Un número a es divisor de un número b si la división de b entre a es exacta. VAMOS A PRACTICAR
  12. 13. CRITERIOS DE DIVISIBILIDAD Nos permiten descubrir si un número es divisible por otro sin necesidad de hacer la división.
  13. 17. Un número es divisible por 11 cuando la diferencia entre la suma de los Valores absolutos de sus cifras de lugar impar y la suma de los valores Absolutos de sus cifras de lugar par, de derecha a izquierda,es 0 o Múltiplo de 11.
  14. 18. VAMOS A PRACTICAR
  15. 21. Para factorizar un número o descomponerlo en factores efectuamos sucesivas divisiones entre sus divisores primos hasta obtener un uno como cociente . Para realizar las divisiones utilizaremos una barra vertical , a la derecha escribimos los divisores primos y a la izquierda los cocientes . DESCOMPOSICIÓN FACTORIAL DE UN NÚMERO La descomposición factorial de 432 es:
  16. 22. Vamos a practicar
  17. 23. Cálculo de los divisores de un número Un número a es divisor de un número b si la división de b entre a es exacta. Para calcular todos los divisores de un número , dividimos dicho número entre los números naturales, es decir, entre 1, 2, 3, ... hasta llegar a la división en la que el cociente sea menor que el divisor. De cada división exacta obtenemos dos divisores : el divisor y el cociente. El conjunto de los divisores de un número a se escribe Div( a ).
  18. 24. Veamos un ejemplo
  19. 25. Vamos a practicar
  20. 26. OTRA FORMA DE CALCULAR LOS DIVISORES DE UN NÚMERO
  21. 27. VEAMOS UN EJEMPLO Div(36)= 1,2,3, 4, 6, 9,12, 18, 36
  22. 28. MÁXIMO COMÚN DIVISOR Para obtener el máximo común divisor de dos números naturales, por ejemplo de 12 y 8, seguimos los siguientes pasos: 1. Hallamos los divisores de uno de los números, por ejemplo del 12; para ello lo dividimos entre todos los números naturales comprendidos entre 1 y 12, ambos incluidos: Los divisores de 12 son aquellos que al dividir han dado resto cero, es decir: 1, 2, 3, 4, 6 y 12. Div (12) = 1,2,3,4,6,12
  23. 29. 2. Hallamos los divisores del otro número, el 8, dividiéndolo entre todos los números naturales comprendidos entre 1 y 8, ambos incluidos: Los divisores de 8 son aquellos que al dividir han dado resto cero, es decir: 1, 2, 4 y 8. . Div(8) = 1,2,4,8
  24. 30. 3. Comparamos los divisores de ambos números, 12 y 8, y vemos los que tienen en común: 1, 2 y 4. El mayor de ellos es 4. Por tanto: m.c.d. (12, 8) = 4
  25. 31. Otra forma de calcular de m.c.d.
  26. 32. NÚMEROS PRIMOS Y COMPUESTOS Llamamos número primo al que solo tiene dos divisores: él mismo y la unidad. En cambio, un número es compuesto si tiene más de dos divisores. <ul><li>divisores de 14 = 1, 2, 7 y 14; como tiene más de dos divisores, 14 es número compuesto;
  27. 33. divisores de 12 = 1, 2, 3, 4, 6 y 12; como tiene más de dos divisores, 12 es número compuesto;
  28. 34. divisores de 7 = 1 y 7; como solo tiene dos divisores (él mismo y la unidad), 7 es número primo. </li></ul>El número 1 no se considera número primo, ya que solo tiene un divisor: él mismo. .
  29. 36. VAMOS A PRACTICAR
  30. 37. fin

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