# Divisibility Proof (Abstract Algebra)

## Homework Statement

let a belong to N and x,r belong to Z use the definition of divisibility along with the axioms of Integers to prove that IF 5|a and 15|(2ax+r) then 5|r

## Homework Equations

How do I continue the proof??

## The Attempt at a Solution

So I have: let a belong to N and x,r belong to Z. Assume 5|a and 15|(2ax+r). Then there is s,t belonging to Z such that a=5s and (2ax+r)=3(5T).

jambaugh
Gold Member
How does the definition apply to 5|r? Can you get the form required to show it?

It is an if then proof with 5|r being what you are trying to prove.

Use the fact that 5|a to substitute.

jambaugh
Gold Member
It is an if then proof with 5|r being what you are trying to prove.

Yes of course but you must end up with your next to last step being the defining form of 5|r and the last step being "thus 5|r".

You almost have it. Play with what you have and what you need and see if you can connect the two. I can tell you but the point of you going through the proof is you going through the proof.

I now have this so
r=15t-2ax=
15t-2(5s)x=
5(3T)-5(2s)x=
5(3t-2sx)=r
since 3t-2sx belongs to Z we have 5|r

does that make sense??

I now have this so
r=15t-2ax=
15t-2(5s)x=
5(3T)-5(2s)x=
5(3t-2sx)=r
since 3t-2sx belongs to Z we have 5|r

does that make sense??

Looks good.