Question

Prove:
If a|b and a|c , then a|sb + tc for all s, t in Z
Please, help.

Answer 1

ok ill help you :P

Answer 2

ok, it's easy, right? wow....

Answer 3

put it in neat, my friend. hihihi

Answer 4

its means
If a|b and a|c , then a devides any linear compination btw b and c

i have two proof , one is easy ill use mod in it and the other is sort of basic using well ordering principle
which wanna you wana me to go through with ?

Answer 5

It's up to you, my saver.:)

Answer 6

ok ill give you both :P

Answer 7

I give you mine, too , hihihi

Answer 8

a| b --> a|sb --> sb = aq
a|c --> a|tc --> tc = ak
--------------------------
sb + tc = aq+ ak = a(q+k) which shows a|(sb + tc)
Am I right? check my stuff, boss!!

Answer 9

yes ! ur correct
this is the first proof ^^

Answer 10

in other way you could say( same proof as urs )
a|b ---> b=0 mod a ----> sb=0 mod a
a|c ---> c=0 mod a ----> tc=0 mod a

then sb+tc=0+0 mod a
sb+tc=0 mod a

Answer 11

Got you!! @ikram

Answer 12

Since not many one willing to help me, can I ask you another question here without posting a new one??

Answer 13

sure , ask anything
ill help if its easy ":P

Answer 14

Prove it here, my friend.

Answer 15

haha ok let me check

Answer 16

ok so this is the quesion ?
How to prove there are only 6 symmetries of an equilateral triangle?

and this is your prof work ?

This is what I get so far: Let S ={A, B, C} where A, B, C are vertices of the triangle. symmetric function f : {A,B,C} --> {A,B,C} has at most 6 possibilities outcome. But I don't know how to make it in logic.

Answer 17

Yes,

Answer 18

Let S ={A, B, C} where A, B, C are vertices of the triangle.


so they would be 3 rotations right ?
im trying to use algebra :o

Answer 19

yes,

Answer 20

we have 3 vertices define a line from vertice to the oposite line
and set identity rotation \( R_{ 360 }\) and \( R_{ 180} \) around each line

you would have 3 subsets of order 2 (2 symetric rotations )
symetric around line from vertics A \(\large S_1=\{R_{ 360A}, R_{ 180A} \}\)
symetric around line from vertics B \(\large S_2=\{R_{ 360B}, R_{ 180B} \}\)
symetric around line from vertics C \(\large S_3=\{R_{ 360C}, R_{ 180C} \}\)

then the union set of symetric would be of order 6
\(\large U_s=\{R_{ 360A},R_{ 360B},R_{ 360C},R_{ 180A},R_{ 180B},R_{ 180C}\}\)

Answer 21

hope its ok :o
good luck

Answer 22

you might see it in other way like this :-

Answer 23

second:-
R120 ,R240 rotations

last :-


so (all semetric would be 1,2,3 )=6