Question 144
Consider addition $\oplus$ and multiplication $\otimes$ modulo 7 of the numbers in $S=\{0,1,2,3,4,5,6\}$. This means that
$$
\begin{aligned}
& m \oplus n=\text{ remainder when } m+n \text{ is divided by } 7 \\
& m \otimes n=\text{ remainder when } m \times n \text{ is divided by } 7 \text{. }
\end{aligned}
$$
Then 1 is the multiplicative identity and each element $a \in S$ has a multiplicative inverse $\frac{1}{a}$. Find the value of $\frac{1}{4} \oplus\left(2 \otimes \frac{1}{3}\right)$ in this number system.
$$
\begin{aligned}
& m \oplus n=\text{ remainder when } m+n \text{ is divided by } 7 \\
& m \otimes n=\text{ remainder when } m \times n \text{ is divided by } 7 \text{. }
\end{aligned}
$$
Then 1 is the multiplicative identity and each element $a \in S$ has a multiplicative inverse $\frac{1}{a}$. Find the value of $\frac{1}{4} \oplus\left(2 \otimes \frac{1}{3}\right)$ in this number system.