if modular inverse is injective or not

Question

Prove that the function $f(x)=(8x+5) \bmod 17$ is injective on the set $ \{0,1,2,3,\cdots,15,16\}$. If so, give its inverse function

I know how to find the inverse of modular function and that will be $\frac{x-5}8 \bmod 17$. I need help in understanding this question.

Answer 1

Hint: In arithmetic modulo $n$, if $\gcd(m, n) = 1$, multiplication by $m$ is injective. (In your case, $\gcd(8, 17) = 1$, so multiplication by $8$ is injective.)

And here is the kicker: There is a number $m'$ such that division by $m$ is the same as multiplication by $m'$. Find that number, and you're basically done.

Answer 2

The question asks you to prove that the function $f$ is injective, that is

if $a,b\in\{0,1,2,\dots,16\}$ and $f(a)=f(b)$, then $a=b$.

When a function from a set to itself is injective, it is also surjective (pigeonhole principle). Thus the function $f$ has an inverse.

Hint for finding the inverse: $2\cdot8=16$.