Question:
For the binary operation \( \times_5 \) on the set \( S = \{1,2,3,4\} \), evaluate:
\[ (3 \times_5 4^{-1})^{-1} \]
Solution:
Step 1: Find inverse of 4 modulo 5
We need \( 4 \times x \equiv 1 \pmod{5} \)
- \(4 \times 4 = 16 \equiv 1 \) ✅
So, \( 4^{-1} = 4 \)
Step 2: Compute \( 3 \times_5 4 \)
\[ 3 \times 4 = 12 \equiv 2 \pmod{5} \]
Step 3: Find inverse of 2 modulo 5
- \(2 \times 3 = 6 \equiv 1 \) ✅
So, \( 2^{-1} = 3 \)
Final Answer:
\[ \boxed{3} \]