Divisibility Proof (Abstract Algebra)

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  • #1
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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).
 

Answers and Replies

  • #2
jambaugh
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How does the definition apply to 5|r? Can you get the form required to show it?
 
  • #3
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It is an if then proof with 5|r being what you are trying to prove.
 
  • #4
Use the fact that 5|a to substitute.
 
  • #5
jambaugh
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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.
 
  • #6
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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??
 
  • #7
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.
 

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