In short, you worked the problem correctly, but there is some confusion about what you are regarding as the "inverse" and "answer." You used a different set of residues, which gives you different numbers from the book, but you and the book still arrived at the same answer, because the answers are congruence classes, not single numbers (note that your inverse is congruent to their inverse, and your solution is congruent to their solution!).
-8 is certainly
an inverse of 13 mod 35, and 27 is
an inverse of 13 mod 35 (note that -8 and 27 are congruent) but neither is
the inverse, and 27 is not more correct than -8. There are many inverses. Namely, any member of the congruence class [27] (which we could also call [-8] with equal validity - they are exactly the same) is an inverse.
The same applies to the value of x. x = 243 and x = -72 are equally valid solutions. Any member of [-72] (or [243] - again, same thing) is a solution. To say that x = 243 is "the" solution is wrong. To say that [243] is "the" solution is correct.
I would expect that your book said that [27] is the inverse, not the number 27. Both the inverse and value for x are entire congruence classes, not single numbers.
Mentallic said:
The numbers modulo 35 range from 0-34, so any number out of that range needs to be moved into it. -8 is out of that range
This is not true. There is nothing special about the set of nonnegative residues, one may use the set of least absolute residues to solve problems (as I think the OP did), and it is often advantageous. One may use any other, really. Using numbers outside of that specific range you chose is not incorrect and can never lead one to an incorrect answer.
When reporting your answers, it is nice to pick a set of residues and report your congruence classes in those terms, but that is purely a matter of notation. If one reports [-8] as the inverse (thereby using least absolute residues), they should report [-2] as the solution. If one reports [27] as the inverse (thereby using least nonnegative residues), they should report [33] as the solution. These are the same answer, though.
