Question 180
A function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ is called *loggy* if it satisfies the following two conditions:
(i) $f(xy) \equiv f(x) + f(y) \pmod{8}$ for all $x, y \in \mathbb{Z}$ that are not divisible by $17$;
(ii) $f(x + 17) \equiv f(x) \pmod{8}$ for all $x \in \mathbb{Z}$.
Determine, with proof,
a. if there exists a loggy function for which $f(2) = 1$;
b. if there exists a loggy function for which $f(3) = 1$.
(i) $f(xy) \equiv f(x) + f(y) \pmod{8}$ for all $x, y \in \mathbb{Z}$ that are not divisible by $17$;
(ii) $f(x + 17) \equiv f(x) \pmod{8}$ for all $x \in \mathbb{Z}$.
Determine, with proof,
a. if there exists a loggy function for which $f(2) = 1$;
b. if there exists a loggy function for which $f(3) = 1$.