Question

a, b, c, x, y are integers. Prove if a l (bx+cy) then alb and alc.

Answer 1

use the fact that
ak=bx+cy for some integer k
to prove ai=b and at=c for integers i and t

Answer 2

but i guess you already knew that
you are looking for an actual hint probably
right?

Answer 3

hmmm.. are you sure this is a true statment
Does it say prove if true? Or provide a counterexample if false?

Answer 4

I think you may want to see if you can find a counterexample instead of a proof.


Now we can prove the converse of that statement.
If a|b and a|c then a|(bx+cy).

ai=b and ak=c
so xb+yc=x(ai)+y(ak)=a(xi+yk)
therefore a|(xb+yc).
Again you can prove the converse.

But I think the statement you are proving is actually false and think your job is to find a counterexample.

Answer 5

It just says prove, so I'm pretty sure it's true.

Answer 6

i'm pretty sure it isn't
because i can think of a counterexample:
I will choose b,x,y, and c
let b=3,x=2,y=4,c=5
so bx+cy=3(2)+4(5)=26=ai
a and i are both integers where I choose a to be 13 and i to be 2.
But a does not divde b and a does not divide c
since 13k=3 where k is integer is not a true statement
also 13j=5 where k is integer is also not a true statement

Answer 7

Hmmm.. ok, this is a 2 part question, maybe I got something wrong in the first part. I'll write the question word for word.

For all integers a, b, and c, if alb and alc, then for all integers x and y, a l (bx + cy).

a) State the converse
b) Prove the converse

Is the converse not what I have above?

Answer 8

if p then q
and the converse of that is
if q then p
you found the converse

Answer 9

I don't see anything wrong with my counterexample...
do you?

Answer 10

let me think about it again i guess

Answer 11

So to confirm, my converse is right? I don't see anything wrong wiht your counterexample either.

Answer 12

I wonder if the orginial is suppose to say:
If for all integers a,b,c if a|b and a|c, there there exists x and y such that a|(bx+cy).
The converse would be.
If there exists integers x and y for all a,b and c such that a|(bx+cy), then a|b and a|c.

This might be true...

Answer 13

but it says a whole bunch of for all's

Answer 14

Or mean If for a,b, and c there exists x and y such that a|(bx+cy), then a|b and a|c. *

Answer 15

because we could choose x and y such that the statement is true

Answer 16

like if x=0 and y=1
then we have
a|c
or if we choose x=1 and y=0
then we have a|b

Answer 17

I'll have to ask someone about it tomorrow... thanks for your help

Answer 18

np
i tried
good luck to you