What is the smallest number that must be added to 13851 so that the resulting number is exactly divisible by 87?

Introduction / Context:
This question tests divisibility and remainders, specifically how to adjust a given number by the smallest amount so that it becomes exactly divisible by a given divisor. It is a standard type of problem in the number system and modular arithmetic section of aptitude exams.

Given Data / Assumptions:
- Original number: 13851.
- Required divisor: 87.
- We must find the least non negative integer k such that 13851 + k is divisible by 87.

Concept / Approach:
If a number N leaves a remainder r when divided by a divisor d, then the smallest number to be added so that N becomes divisible by d is (d − r), provided r is not zero. We therefore need to compute the remainder when 13851 is divided by 87 and adjust accordingly.

Step-by-Step Solution:
Compute 13851 ÷ 87 and find the remainder r. Performing the division, 87 × 159 = 13833. Subtract: 13851 − 13833 = 18, so the remainder r is 18. The smallest number to be added is 87 − 18 = 69. Therefore, 13851 + 69 = 13920, and 13920 ÷ 87 is an exact division.

Verification / Alternative Check:
Check directly: 13920 ÷ 87 = 160 exactly, with no remainder. If we added any smaller positive number, such as 54 or 43, the result would still not be divisible by 87 because the remainder would not be driven to zero. Hence, 69 is indeed the minimum such addition.

Why Other Options Are Wrong:
Adding 18 leads to 13869, which divided by 87 still leaves a remainder, since 87 × 159 = 13833 and 13869 − 13833 = 36.
Adding 43 gives 13851 + 43 = 13894, which is not divisible by 87.
Adding 54 gives 13905, which again is not an exact multiple of 87.
Adding 33 is arbitrary and does not result in a multiple of 87 either.

Common Pitfalls:
A frequent error is to miscalculate the remainder or to forget that we need the smallest non negative adjustment. Some students also confuse the operation and subtract instead of adding, or they directly test options without understanding the relationship N + k ≡ 0 (mod d). Practising careful long division or modular arithmetic helps avoid such mistakes.

Final Answer:
The smallest number that must be added to 13851 to make it divisible by 87 is 69.