Question:
Write the multiplication table for the set of integers modulo 5, i.e., \( Z_5 = \{0,1,2,3,4\} \).
Concept:
In modulo 5 multiplication,
\[ a \times_5 b = (a \times b) \mod 5 \]
Each product is reduced to its remainder when divided by 5.
Solution:
Step 1: Compute products and reduce modulo 5.
- \( 2 \times 3 = 6 \equiv 1 \pmod{5} \)
- \( 4 \times 4 = 16 \equiv 1 \pmod{5} \)
- \( 3 \times 4 = 12 \equiv 2 \pmod{5} \)
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 multiplication table (Cayley table) for integers modulo 5.