Thread: Divides proof

I'm struggling with the following question:

let a, b, c, k, l be integers
Prove that if: a|b and a|c then a|(kb + lc)

I'm not looking for the answer outright just a little help.

I realise that:
b = ka
c = la

(kb + lc) = m x a

The approach I've been taking so far is to go: b + c = ka + kl and then try to rearrange (ka + kl) to (kb + lc) since I know that
if a|b and a|c then a|(b + c)

Am I on the right track? If so then I know it's my algebra skills which are the problem.

and are arbitrary integers. It doesn't make sense to let and . Plus it confuses readers like me....

You can use different integers, e.g. and , in which , clearly, divides this.

An alternate solution is to use mods, which works just as well (and it doesn't need random variables!).

Thanks for that Richard, makes much more sense.

We have been studying mods lately so out of interest, how would you approach this using mods?

Basically, and , so for any integers and ,




Therefore the sum is also congruent to 0 (mod a).