The Stolen Sapphire

Room 51 is the primary answer (smallest valid candidate). Both 51 and 82 satisfy all constraints: multiple of 17, not prime, and n² ≡ 4 (mod 7). Room 34 fails the modular test (34² mod 7 = 1). Room 17 is eliminated as prime. Inspector Varsha's key technique: compute n mod 7 first, then square it.

Work through each constraint systematically for all four candidates. Modular arithmetic on squares is efficiently done by first reducing the base mod 7, then squaring the result. Step 1: Candidates: 17, 34, 51, 82. All are multiples of 17 ✓. Step 2: Primality check: 17 is prime → eliminate. 34 = 2×17 (not prime ✓). 51 = 3×17 (not prime ✓). 82 = 2×41 (not prime ✓). Remaining: 34, 51, 82. Step 3: Compute n² mod 7 for each: use (n mod 7)² mod 7. Step 4: 34 mod 7 = 34 − 4×7 = 34 − 28 = 6. 6² = 36. 36 mod 7 = 36 − 5×7 = 1. So 34² mod 7 = 1 ≠ 4. Eliminate 34. Step 5: 51 mod 7 = 51 − 7×7 = 51 − 49 = 2. 2² = 4. 4 mod 7 = 4. So 51² mod 7 = 4 ✓. Step 6: 82 mod 7 = 82 − 11×7 = 82 − 77 = 5. 5² = 25. 25 mod 7 = 25 − 3×7 = 4. So 82² mod 7 = 4 ✓. Both 51 and 82 satisfy all stated constraints. Key Insight: Computing n² mod m directly for large n is tedious. The shortcut is always to first reduce n mod m, then square the result and reduce again. This is because (n mod m)² mod m = n² mod m — modular arithmetic distributes over multiplication. This technique appears constantly in cryptography and competitive programming.