Question:
Write the composition table for multiplication modulo 10 (\( \times_{10} \)) on the set \( S = \{2,4,6,8\} \).
Concept:
The operation is defined as:
\[ a \times_{10} b = (a \times b) \mod 10 \]
Solution:
Step 1: Compute values
- \(2 \times 2 = 4 \equiv 4\)
- \(2 \times 4 = 8 \equiv 8\)
- \(2 \times 6 = 12 \equiv 2\)
- \(2 \times 8 = 16 \equiv 6\)
- \(4 \times 4 = 16 \equiv 6\)
- \(4 \times 6 = 24 \equiv 4\)
- \(4 \times 8 = 32 \equiv 2\)
- \(6 \times 6 = 36 \equiv 6\)
- \(6 \times 8 = 48 \equiv 8\)
- \(8 \times 8 = 64 \equiv 4\)
Step 2: Construct the table
\[ \begin{array}{c|cccc} \times_{10} & 2 & 4 & 6 & 8 \\ \hline 2 & 4 & 8 & 2 & 6 \\ 4 & 8 & 6 & 4 & 2 \\ 6 & 2 & 4 & 6 & 8 \\ 8 & 6 & 2 & 8 & 4 \\ \end{array} \]
Final Answer:
The above table is the required composition table for multiplication modulo 10 on the set \( \{2,4,6,8\} \).