Hello,

Yes.Here's what I have:

257 = 198 x 1 + 59 (1)

198 = 59 x 3 + 21 (2)

59 = 21 x 2 + 17 (3)

21 = 17 x 1 + 4 (4)

17 = 4 x 4 + 1 (5)

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=5x(59-21x2)-4x21

1=5x59-10x21-4x21

1=5x59-14x21

Then from (2), we have 21=198-59x3

This makes :

1=5x59-14x(198-59x3)

1=5x59-14x198+59x52

1=57x59-14x198

Last step : from (1), we have 59=257-198

1=57x(257-198)-14x198

1=57x257-57x198-14x198

1=57x257-71x198

So -71x198=1-57x257

-71x198=1 mod 257

find what -71 mod 257 is and you have your answer.

for the () stuff, you just have to keep an eye over the successive remainders. (they're in color above)