I can't seem to make Euclidean work backwards either. :/

Use the extended Euclidean algorithm to find 198^-1 mod 257.

Here's what I have:

257 = 198 x 1 + 59

198 = 59 x 3 + 21

59 = 21 x 2 + 17

21 = 17 x 1 + 4

17 = 4 x 4 + 1

Backwards:

1 = 17 - 4 x 4

1 = 17 - 4 (21 - 17)

1 = 5 x 17 - 4 x 21

1 = 5 x (59 - 21 x 2) - 4 x 21

I worked a little more and think I got this:

1 = 5 x 59 - 14 x 21 (this was by trial and error of some sort)

1 = 5 x 59 - 14 x (198 - 59 x 3)

And I'm stuck again.

And that's where I'm stuck. I'm pretty sure it's having multiplication in the () that's screwing me up, not sure I worked with it before.. and if I have, I just don't get it.