Abstract Alg. Proof w/ Mod Congruence and Relative Primes

[Colleen G]

Homework Statement

If ≡(mod), ≡(mod),and gcd(,)=1,provethat ≡ (mod ).

Homework Equations

If ≡(mod)→n|ab-cd
≡(mod)→n|b-d
gcd(,)=1→ relatively prime. So bx+ny=1

Need to show n|a-c→a-c=nw

The Attempt at a Solution

If n|ab-cd, then nk=ab-cd
If n|b-d, then nl=b-d
If n|ab-cd AND n|b-d, then n|p(ab-cd)+q(b-d). So pab-pcd+qb-qd.
pad-pcd+qb-qd
=pab+qb-pcd-qd
=b(pa+q) +d(-pc-q)

I'm stuck! Don't know if this is going anywhere.

 

[Stephen Tashi]

If ≡(mod), ≡(mod),and gcd(,)=1,provethat ≡ (mod ).

You need to edit your post so the variables show up. ( https://www.physicsforums.com/help/latexhelp/ )
Do you mean something like ab \equiv cd \ (mod \ n) ?