Discrete Math - a modulus proof

[CaptainSFS]

Homework Statement

I have to prove the following claim.

Claim: For any positive integers m and n, m and n both greater than 1, if n|m and a≡b(mod m), then a≡b(mod n).

Homework Equations

n/a

The Attempt at a Solution

so i first changed each equation (ex: a≡b(mod m)) to a=b+qm and a=b+qn

I figured in these forms I could show that the equations are equal.

so I eventually get (a-b)/q=m or =n respectively. So I believe this shows their equality, but i am completely unsure because it won't always work I don't think. I need to also show that n|m. So tired dividing the m=(a-b)/q by the n= equation and of course I just get 1...

To be completely honest I am not quite sure how to prove this. I am not quite familiar with the mod function and I am incredibly weak with proofs. If anyone can give me insight into solving this problem I would great appreciative.

also note that this should be able to be done with a direct proof.

thanks!

 

[Dick]

a=b(mod m) means m|(a-b). a=b(mod n) means n|(a-b). If n|m what can you conclude from this?

 

[CaptainSFS]

I guess you could conclude that (m|(a-b)) / (n|(a-b)). Or I guess that's like m|n? Actually I'm not really sure. I'm not sure if that's correct. If it is, I'm not really sure if that proves the statement. If m|n is true, does that mean they're the same integer? In which case I assume that would prove it.

 

[Dick]

m|(a-b) and n|(a-b) are true/false statements. You can't DIVIDE them. Think about this. If a|b and b|c then a|c, right?

 

[CaptainSFS]

m|(a-b) and n|(a-b) are true/false statements. You can't DIVIDE them. Think about this. If a|b and b|c then a|c, right?

Right, i mean that makes sense, but that's just reiterating the claim isn't it?

If n|m and m|(a-b), then n|(a-b) is like saying If a|b and b|c then a|c.

I think that there is some constant k so that kn=m? Then in this case, because they're modular, they come up with the same answer. Is that any more correct?

 

[Dick]

Yes. I just really didn't like (m|(a-b)) / (n|(a-b)). Didn't make much sense to me. Maybe it did to you.

 

[CaptainSFS]

I see, but this is where my weak point is. I'm not sure how I write a proof for that.

 

[Dick]

"If n|m and m|(a-b), then n|(a-b) is like saying If a|b and b|c then a|c." That's the basis of the proof. Can you just flesh that out into a full proof? Start with "if n|m and a≡b(mod m)" and conclude with "then a≡b(mod n)".

 

[HallsofIvy]

If a= b mod m, then m divides a- b so a-b= km for some integer k. If n divides m then m= pn for some integer p. Put those together to get that a-b= ?n.

 

[CaptainSFS]

alright, thanks you two. Hopefully that'll help. I'll let you know if I have any further questions. Thanks! :)