Question:
Find the inverse of \(5\) under multiplication modulo \(11\) on \(Z_{11}\).
Concept:
The inverse of an element \(a\) modulo \(11\) is a number \(b\) such that:
\[ a \times b \equiv 1 \pmod{11} \]
Solution:
Step 1: Find \(b\) such that
\[ 5 \times b \equiv 1 \pmod{11} \]
Step 2: Try values from \(Z_{11} = \{0,1,2,\dots,10\}\)
- \(5 \times 1 = 5 \equiv 5\)
- \(5 \times 2 = 10 \equiv 10\)
- \(5 \times 3 = 15 \equiv 4\)
- \(5 \times 4 = 20 \equiv 9\)
- \(5 \times 5 = 25 \equiv 3\)
- \(5 \times 6 = 30 \equiv 8\)
- \(5 \times 7 = 35 \equiv 2\)
- \(5 \times 8 = 40 \equiv 7\)
- \(5 \times 9 = 45 \equiv 1\) ✅
Step 3: Therefore,
\[ 5 \times 9 \equiv 1 \pmod{11} \]
Final Answer:
\[ \boxed{9} \]