Unifying Contradiction

I hate the damn things. I know just enough number theory that I know when one would be useful, and then I take forever to calculate one. And Mathematica was of little help. I had to scour some web sites to determine that it uses a weird variant of its Solve function for linear congruence relations.

Here I was trying to get it to Solve[ Mod[17 a, 10000] == b, a ], which is the normal syntax (and which doesn't work) and it turns out I need to say Solve[ 17 a == b && Modulus == 10000, a], a fact which is not mentioned in the reference for EITHER Solve or Mod.