Question:
Construct the composition table for \( \times_5 \) on \( Z_5 = \{0,1,2,3,4\} \).
Concept:
In modular arithmetic,
\[ a \times_5 b = (a \times b) \mod 5 \]
i.e., multiply two numbers and take the remainder after division by 5. :contentReference[oaicite:0]{index=0}
Solution:
Step 1: Compute values using modulo 5.
- \( 2 \times_5 3 = 6 \mod 5 = 1 \)
- \( 3 \times_5 4 = 12 \mod 5 = 2 \)
- \( 4 \times_5 4 = 16 \mod 5 = 1 \)
Step 2: Construct the table
\[ \begin{array}{c|ccccc} \times_5 & 0 & 1 & 2 & 3 & 4 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 \\ 2 & 0 & 2 & 4 & 1 & 3 \\ 3 & 0 & 3 & 1 & 4 & 2 \\ 4 & 0 & 4 & 3 & 2 & 1 \\ \end{array} \]
Final Answer:
The above table is the required composition (Cayley) table for multiplication modulo 5 on \( Z_5 \). :contentReference[oaicite:1]{index=1}