Congruence problem with gcd
1. Let d,k,n be integers with n>d>1. If d=gcd(k,n), prove there exist an integer e such that 1<e<n and ke congruence 0 (mod n).
My proof so far:
Now d > 1 implies d >= 2, n > d implies n>=3, therefore there exists a integer e such that n > e > 1 with e >=2.
Now I understand that I need to have something like ke = 0 + na for an integer a for the second part of the proof, now I seem to figure out.
I wrote stuff like k = dv and d = ki + nj for integer v,i,j, then I have ke = dve = kvi + njve, but now I'm stuck.
Am I doing the right thing?