Smallest number with specified remainders: Find the least positive number that leaves remainders 3, 5, and 8 when divided by 16, 18, and 21 respectively.

Introduction / Context:This is a Chinese Remainder style problem with a useful observation: the three remainders (3, 5, 8) are all congruent to −13 with respect to their moduli (16, 18, 21). That means the sought number differs by the same constant from a common multiple of the moduli.

Given Data / Assumptions:

• N ≡ 3 (mod 16)
• N ≡ 5 (mod 18)
• N ≡ 8 (mod 21)

Concept / Approach:Note that 3 = −13 mod 16, 5 = −13 mod 18, 8 = −13 mod 21. Therefore, N ≡ −13 modulo LCM(16, 18, 21). Compute the LCM and then find the least positive N of the form LCM − 13.

Step-by-Step Solution:

Verification / Alternative check:Check: 995 mod 16 = 3, 995 mod 18 = 5, 995 mod 21 = 8. All conditions satisfied.

Why Other Options Are Wrong:893, 992, 1024, and 1005 do not satisfy all three remainder conditions simultaneously.

Common Pitfalls:Attempting to solve each congruence independently without noticing the convenient common shift of −13, or computing LCM incorrectly.

Final Answer:995