The River Island Census

23 people live on the island. Verified: 23 = 3×7+2 (remainder 2 when divided by 3), 23 = 5×4+3 (remainder 3 when divided by 5), 23 = 7×3+2 (remainder 2 when divided by 7). The next solution, 128, exceeds 100.

The Chinese Remainder Theorem guarantees a unique solution modulo (3×5×7=105) when the moduli are pairwise coprime — which 3, 5, and 7 are. The unique solution in the range 1-105 is found by systematic search or algebraic construction. Step 1: Translate clues: N ≡ 2 (mod 3) means N leaves remainder 2 when divided by 3. N ≡ 3 (mod 5) means remainder 3 when divided by 5. N ≡ 2 (mod 7) means remainder 2 when divided by 7. Step 2: Start with N ≡ 2 (mod 7): candidates are 2, 9, 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, 93. Step 3: Filter for N ≡ 2 (mod 3): check which of the above leave remainder 2 when divided by 3. 2 mod 3 = 2 ✓. 9 mod 3 = 0 ✗. 16 mod 3 = 1 ✗. 23 mod 3 = 2 ✓. 30 mod 3 = 0 ✗. 37 mod 3 = 1 ✗. 44 mod 3 = 2 ✓. Remaining candidates: 2, 23, 44, 65, 86. Step 4: Filter for N ≡ 3 (mod 5): 2 mod 5 = 2 ✗. 23 mod 5 = 3 ✓. 44 mod 5 = 4 ✗. 65 mod 5 = 0 ✗. 86 mod 5 = 1 ✗. Step 5: Only N = 23 satisfies all three conditions and is less than 100. Verify: 23 = 7×3+2 ✓. 23 = 5×4+3 ✓. 23 = 3×7+2 ✓. N = 23. Step 6: The next solution would be 23 + 105 = 128, which exceeds 100 — confirming uniqueness below 100. Key Insight: The CRT tells us that when moduli are pairwise coprime, there is always a unique solution within the product of all moduli (here 3×5×7=105). The 'less than 100' constraint was redundant in this case — but it elegantly narrows the puzzle to a single human-scale answer.