Inputs [...] must be equal? No, that's not right. Phrasing it in English is a nice exercise, so I recommend trying again.

You mean, e.g.

Isn't b mod 17 = d mod 17 what you were supposed to prove, i.e., isn't it the end of the proof of injectivity? What you are writing is correct, though I am not sure you you showed why the facts thataand 17 are relatively prime and a*b mod 17 = a*d mod 17 imply that b mod 17 = d mod 17, i.e., whyacan be canceled.

Any cycle (x_{1}, x_{2}, ..., x_{n}) equals the composition of n - 1 transpositions (x_{1}x_{2})(x_{2}x_{3}) ... (x_{n-1}x_{n}). Therefore, a cycle of length n is even iff n is odd.