Para obtener los distintos criterios de divisibilidad se utilizan las congruencias, con esto se obtienen los restos potenciales que servirán para sacar la expresión del criterio.
Criterio de divisibilidad del 2
editar
10
0
≡
1
(
mod
2
)
{\displaystyle 10^{0}\equiv 1{\pmod {2}}}
10
1
≡
0
(
mod
2
)
{\displaystyle 10^{1}\equiv 0{\pmod {2}}}
10
2
≡
0
(
mod
2
)
{\displaystyle 10^{2}\equiv 0{\pmod {2}}}
10
3
≡
0
(
mod
2
)
{\displaystyle 10^{3}\equiv 0{\pmod {2}}}
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
10
k
−
1
≡
0
(
mod
2
)
{\displaystyle 10^{k-1}\equiv 0{\pmod {2}}}
10
k
≡
0
(
mod
2
)
{\displaystyle 10^{k}\equiv 0{\pmod {2}}}
El criterio de divisibilidad del 2 es 2/4/6/8/10/12/14/16/18/20
a
≡
a
k
⋅
0
+
a
k
−
1
⋅
0
+
.
.
.
+
a
1
⋅
0
+
a
0
⋅
1
(
mod
2
)
{\displaystyle a\equiv a_{k}\cdot 0+a_{k-1}\cdot 0+...+a_{1}\cdot 0+a_{0}\cdot 1{\pmod {2}}}
Observando las el desarrollo de las congruencias se puede llegar a la conclusión de que el único resto potencial distinto de 0 es
r
0
{\displaystyle r_{0}}
a
0
{\displaystyle a_{0}}
Todos los números pares cumplen el criterio de divisibilidad del 2.
Criterio de divisibilidad del 3
editar
10
0
≡
1
(
mod
3
)
{\displaystyle 10^{0}\equiv 1{\pmod {3}}}
10
1
≡
1
(
mod
3
)
{\displaystyle 10^{1}\equiv 1{\pmod {3}}}
10
2
≡
1
(
mod
3
)
{\displaystyle 10^{2}\equiv 1{\pmod {3}}}
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
10
k
−
1
≡
1
(
mod
3
)
{\displaystyle 10^{k-1}\equiv 1{\pmod {3}}}
10
k
≡
1
(
mod
3
)
{\displaystyle 10^{k}\equiv 1{\pmod {3}}}
a
≡
a
k
⋅
1
+
a
k
−
1
⋅
1
+
.
.
.
+
a
1
⋅
1
+
a
0
⋅
1
(
mod
3
)
{\displaystyle a\equiv a_{k}\cdot 1+a_{k-1}\cdot 1+...+a_{1}\cdot 1+a_{0}\cdot 1{\pmod {3}}}
Si la suma de los dígitos de un número es divisible por 3 entonces ese número será también divisible.
Criterio de divisibilidad del 4
editar
10
0
≡
1
(
mod
4
)
{\displaystyle 10^{0}\equiv 1{\pmod {4}}}
10
1
≡
2
(
mod
4
)
{\displaystyle 10^{1}\equiv 2{\pmod {4}}}
10
2
≡
0
(
mod
4
)
{\displaystyle 10^{2}\equiv 0{\pmod {4}}}
10
3
≡
0
(
mod
4
)
{\displaystyle 10^{3}\equiv 0{\pmod {4}}}
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
10
k
−
1
≡
0
(
mod
4
)
{\displaystyle 10^{k-1}\equiv 0{\pmod {4}}}
10
k
≡
0
(
mod
4
)
{\displaystyle 10^{k}\equiv 0{\pmod {4}}}
El criterio de divisibilidad del 4 es
a
≡
a
k
⋅
0
+
a
k
−
1
⋅
0
+
.
.
.
+
a
1
⋅
2
+
a
0
⋅
1
(
mod
4
)
{\displaystyle a\equiv a_{k}\cdot 0+a_{k-1}\cdot 0+...+a_{1}\cdot 2+a_{0}\cdot 1{\pmod {4}}}
Sus dos últimas cifras tienen que ser divisible por 4 para que el número lo sea.
Criterio de divisibilidad del 5
editar
10
0
≡
1
(
mod
5
)
{\displaystyle 10^{0}\equiv 1{\pmod {5}}}
10
1
≡
0
(
mod
5
)
{\displaystyle 10^{1}\equiv 0{\pmod {5}}}
10
2
≡
0
(
mod
5
)
{\displaystyle 10^{2}\equiv 0{\pmod {5}}}
10
3
≡
0
(
mod
5
)
{\displaystyle 10^{3}\equiv 0{\pmod {5}}}
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
10
k
−
1
≡
0
(
mod
5
)
{\displaystyle 10^{k-1}\equiv 0{\pmod {5}}}
10
k
≡
0
(
mod
5
)
{\displaystyle 10^{k}\equiv 0{\pmod {5}}}
El criterio de divisibilidad del 5 es
a
≡
a
k
⋅
0
+
a
k
−
1
⋅
0
+
.
.
.
+
a
1
⋅
0
+
a
0
⋅
1
(
mod
5
)
{\displaystyle a\equiv a_{k}\cdot 0+a_{k-1}\cdot 0+...+a_{1}\cdot 0+a_{0}\cdot 1{\pmod {5}}}
Si el número termina en 0 o 5 es divisible.
Criterio de divisibilidad del 6
editar
10
0
≡
1
(
mod
6
)
{\displaystyle 10^{0}\equiv 1{\pmod {6}}}
10
1
≡
4
(
mod
6
)
{\displaystyle 10^{1}\equiv 4{\pmod {6}}}
10
2
≡
4
(
mod
6
)
{\displaystyle 10^{2}\equiv 4{\pmod {6}}}
10
3
≡
4
(
mod
6
)
{\displaystyle 10^{3}\equiv 4{\pmod {6}}}
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
10
k
−
1
≡
4
(
mod
6
)
{\displaystyle 10^{k-1}\equiv 4{\pmod {6}}}
10
k
≡
4
(
mod
6
)
{\displaystyle 10^{k}\equiv 4{\pmod {6}}}
El criterio de divisibilidad del 6 es
a
≡
a
k
⋅
4
+
a
k
−
1
⋅
4
+
.
.
.
+
a
1
⋅
4
+
a
0
⋅
1
(
mod
6
)
{\displaystyle a\equiv a_{k}\cdot 4+a_{k-1}\cdot 4+...+a_{1}\cdot 4+a_{0}\cdot 1{\pmod {6}}}
Un número es divisible por 6 si se cumple el criterio de divisibilidad del 2 y a la vez el del 3.
Criterio de divisibilidad del 7
editar
10
0
≡
1
(
mod
7
)
{\displaystyle 10^{0}\equiv 1{\pmod {7}}}
10
1
≡
3
(
mod
7
)
{\displaystyle 10^{1}\equiv 3{\pmod {7}}}
10
2
≡
2
(
mod
7
)
{\displaystyle 10^{2}\equiv 2{\pmod {7}}}
10
3
≡
6
(
mod
7
)
{\displaystyle 10^{3}\equiv 6{\pmod {7}}}
10
4
≡
4
(
mod
7
)
{\displaystyle 10^{4}\equiv 4{\pmod {7}}}
10
5
≡
5
(
mod
7
)
{\displaystyle 10^{5}\equiv 5{\pmod {7}}}
10
6
≡
1
(
mod
7
)
{\displaystyle 10^{6}\equiv 1{\pmod {7}}}
10
7
≡
3
(
mod
7
)
{\displaystyle 10^{7}\equiv 3{\pmod {7}}}
Cada seis cifras se observa una repetición de los restos potenciales.
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
10
k
−
5
≡
1
(
mod
7
)
{\displaystyle 10^{k-5}\equiv 1{\pmod {7}}}
10
k
−
4
≡
3
(
mod
7
)
{\displaystyle 10^{k-4}\equiv 3{\pmod {7}}}
10
k
−
3
≡
2
(
mod
7
)
{\displaystyle 10^{k-3}\equiv 2{\pmod {7}}}
10
k
−
2
≡
6
(
mod
7
)
{\displaystyle 10^{k-2}\equiv 6{\pmod {7}}}
10
k
−
1
≡
4
(
mod
7
)
{\displaystyle 10^{k-1}\equiv 4{\pmod {7}}}
10
k
≡
5
(
mod
7
)
{\displaystyle 10^{k}\equiv 5{\pmod {7}}}
El criterio de divisibilidad del 7 es
a
≡
a
k
⋅
5
+
a
k
−
1
⋅
4
+
a
k
−
2
⋅
6
+
a
k
−
3
⋅
2
+
a
k
−
4
⋅
3
+
a
k
−
5
⋅
1
+
.
.
.
+
a
7
⋅
3
+
a
6
⋅
1
+
a
5
⋅
5
+
a
4
⋅
4
+
a
3
⋅
6
+
a
2
⋅
2
+
a
1
⋅
3
+
a
0
⋅
1
(
mod
7
)
{\displaystyle a\equiv a_{k}\cdot 5+a_{k-1}\cdot 4+a_{k-2}\cdot 6+a_{k-3}\cdot 2+a_{k-4}\cdot 3+a_{k-5}\cdot 1+...+a_{7}\cdot 3+a_{6}\cdot 1+a_{5}\cdot 5+a_{4}\cdot 4+a_{3}\cdot 6+a_{2}\cdot 2+a_{1}\cdot 3+a_{0}\cdot 1{\pmod {7}}}
Criterio de divisibilidad del 9
editar
10
0
≡
1
(
mod
9
)
{\displaystyle 10^{0}\equiv 1{\pmod {9}}}
10
1
≡
1
(
mod
9
)
{\displaystyle 10^{1}\equiv 1{\pmod {9}}}
10
2
≡
1
(
mod
9
)
{\displaystyle 10^{2}\equiv 1{\pmod {9}}}
10
3
≡
1
(
mod
9
)
{\displaystyle 10^{3}\equiv 1{\pmod {9}}}
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
10
k
−
1
≡
1
(
mod
9
)
{\displaystyle 10^{k-1}\equiv 1{\pmod {9}}}
10
k
≡
1
(
mod
9
)
{\displaystyle 10^{k}\equiv 1{\pmod {9}}}
El criterio de divisibilidad del 9 es
a
≡
a
k
⋅
1
+
a
k
−
1
⋅
1
+
.
.
.
+
a
1
⋅
1
+
a
0
⋅
1
(
mod
9
)
{\displaystyle a\equiv a_{k}\cdot 1+a_{k-1}\cdot 1+...+a_{1}\cdot 1+a_{0}\cdot 1{\pmod {9}}}
Un número es divisible por 9 si la suma de sus cifras lo es.
Criterio de divisibilidad del 10
editar
10
0
≡
1
(
mod
10
)
{\displaystyle 10^{0}\equiv 1{\pmod {10}}}
10
1
≡
0
(
mod
10
)
{\displaystyle 10^{1}\equiv 0{\pmod {10}}}
10
2
≡
0
(
mod
10
)
{\displaystyle 10^{2}\equiv 0{\pmod {10}}}
10
3
≡
0
(
mod
10
)
{\displaystyle 10^{3}\equiv 0{\pmod {10}}}
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
10
k
−
1
≡
0
(
mod
10
)
{\displaystyle 10^{k-1}\equiv 0{\pmod {10}}}
10
k
≡
0
(
mod
10
)
{\displaystyle 10^{k}\equiv 0{\pmod {10}}}
El criterio de divisibilidad del 10 es
a
≡
a
k
⋅
1
+
a
k
−
1
⋅
1
+
.
.
.
+
a
1
⋅
2
+
a
0
⋅
1
(
mod
10
)
{\displaystyle a\equiv a_{k}\cdot 1+a_{k-1}\cdot 1+...+a_{1}\cdot 2+a_{0}\cdot 1{\pmod {10}}}
Para que sea divisible por 10 el número tiene que acabar en 0.
Criterio de divisibilidad del 13 en base 28
editar
28
0
=
1
≡
1
(
mod
13
)
{\displaystyle 28^{0}=1\equiv 1{\pmod {13}}}
28
1
=
28
≡
2
(
mod
13
)
{\displaystyle 28^{1}=28\equiv 2{\pmod {13}}}
28
2
=
784
≡
4
(
mod
13
)
{\displaystyle 28^{2}=784\equiv 4{\pmod {13}}}
28
3
=
21952
≡
8
(
mod
13
)
{\displaystyle 28^{3}=21952\equiv 8{\pmod {13}}}
28
4
=
614656
≡
3
(
mod
13
)
{\displaystyle 28^{4}=614656\equiv 3{\pmod {13}}}
28
5
=
17210368
≡
6
(
mod
13
)
{\displaystyle 28^{5}=17210368\equiv 6{\pmod {13}}}
28
6
=
481890304
≡
12
(
mod
13
)
{\displaystyle 28^{6}=481890304\equiv 12{\pmod {13}}}
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
a
≡
a
0
⋅
1
+
a
1
⋅
2
+
a
2
⋅
4
+
a
3
⋅
8
+
a
4
⋅
3
+
a
5
⋅
6
+
a
6
⋅
12
+
.
.
.
(
mod
13
)
{\displaystyle a\equiv a_{0}\cdot 1+a_{1}\cdot 2+a_{2}\cdot 4+a_{3}\cdot 8+a_{4}\cdot 3+a_{5}\cdot 6+a_{6}\cdot 12+...{\pmod {13}}}
Criterio de divisibilidad del 11 en base 13
editar
13
0
=
1
≡
1
(
mod
11
)
{\displaystyle 13^{0}=1\equiv 1{\pmod {11}}}
13
1
=
13
≡
2
(
mod
11
)
{\displaystyle 13^{1}=13\equiv 2{\pmod {11}}}
13
2
=
169
≡
4
(
mod
11
)
{\displaystyle 13^{2}=169\equiv 4{\pmod {11}}}
13
4
=
2197
≡
8
(
mod
11
)
{\displaystyle 13^{4}=2197\equiv 8{\pmod {11}}}
13
4
=
28561
≡
5
(
mod
11
)
{\displaystyle 13^{4}=28561\equiv 5{\pmod {11}}}
13
5
=
371293
≡
10
(
mod
11
)
{\displaystyle 13^{5}=371293\equiv 10{\pmod {11}}}
13
6
=
4826809
≡
9
(
mod
11
)
{\displaystyle 13^{6}=4826809\equiv 9{\pmod {11}}}
13
7
=
6274861
≡
7
(
mod
11
)
{\displaystyle 13^{7}=6274861\equiv 7{\pmod {11}}}
13
8
=
815730721
≡
3
(
mod
11
)
{\displaystyle 13^{8}=815730721\equiv 3{\pmod {11}}}
13
9
=
10604499373
≡
6
(
mod
11
)
{\displaystyle 13^{9}=10604499373\equiv 6{\pmod {11}}}
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
∙
{\displaystyle \bullet }
El criterio de divisibilidad del 11 es
a
≡
a
0
⋅
1
+
a
1
⋅
2
+
a
2
⋅
4
+
a
3
⋅
8
+
a
4
⋅
5
+
a
5
⋅
10
+
a
6
⋅
9
+
a
7
⋅
7
+
a
8
⋅
3
+
a
9
⋅
6
+
.
.
.
(
mod
11
)
{\displaystyle a\equiv a_{0}\cdot 1+a_{1}\cdot 2+a_{2}\cdot 4+a_{3}\cdot 8+a_{4}\cdot 5+a_{5}\cdot 10+a_{6}\cdot 9+a_{7}\cdot 7+a_{8}\cdot 3+a_{9}\cdot 6+...{\pmod {11}}}
Datos: Q97369409