1. **Problem Statement:** We have 5 keys and 5 locks. Each key has a number of engraved lines, and each lock requires a key with specific characteristics based on the number of engraved lines. We need to match each key to its lock, write a mathematical statement connecting keys and locks, display the relationship, and predict the number of engraved lines for Key 10 if the pattern continues. 2. **Matching Keys to Locks:** - Key 1 has 2 lines (smallest even number). - Key 2 has 3 lines (prime number). - Key 3 has 4 lines (less than 5). - Key 5 has 6 lines (more than 4). - Key 4 has 5 lines (exactly 6 lines required for Lock 5, but Key 4 has 5 lines, so this is a mismatch; we must check carefully). Check conditions: - Lock 1 requires smallest even number: Key 1 (2 lines). - Lock 2 requires prime number: Key 2 (3 lines). - Lock 3 requires less than 5 lines: Keys 1 (2), 2 (3), 3 (4), 4 (5), 5 (6) but only Key 3 has 4 lines which is less than 5. - Lock 4 requires more than 4 lines: Keys 4 (5 lines), 5 (6 lines). - Lock 5 requires exactly 6 lines: Key 5 (6 lines). Since Key 4 has 5 lines but Lock 5 requires exactly 6 lines, Key 4 must match Lock 4 (more than 4 lines). Final matching: - Lock 1: Key 1 (2 lines) - Lock 2: Key 2 (3 lines) - Lock 3: Key 3 (4 lines) - Lock 4: Key 4 (5 lines) - Lock 5: Key 5 (6 lines) 3. **Mathematical Statement:** Let $K_n$ be the number of engraved lines on Key $n$. The pattern of engraved lines is: $$K_n = n + 1$$ for $n=1,2,3,4,5$. 4. **Relationship Display:** Keys and locks are paired by the number of engraved lines satisfying lock conditions: - Lock 1: $K_1=2$ (smallest even) - Lock 2: $K_2=3$ (prime) - Lock 3: $K_3=4$ (less than 5) - Lock 4: $K_4=5$ (more than 4) - Lock 5: $K_5=6$ (exactly 6) 5. **Prediction for Key 10:** Using the formula: $$K_{10} = 10 + 1 = 11$$ So, Key 10 would have 11 engraved lines. **Final answers:** - a) Matching: Key 1-Lock 1, Key 2-Lock 2, Key 3-Lock 3, Key 4-Lock 4, Key 5-Lock 5. - b) (i) $K_n = n + 1$ (ii) Relationship as above. (iii) $K_{10} = 11$ lines.