Divisibility adjustment — What is the least number that must be subtracted from 13601 so that the result is exactly divisible by 87? Compute the minimal subtraction using the remainder concept.

Introduction / Context:This problem tests fast divisibility reasoning. Instead of performing long division fully, you can use the idea of remainders: if a number leaves a remainder r when divided by a divisor d, then subtracting r creates a number that is exactly divisible by d. The goal is to find the smallest subtraction that achieves divisibility by 87 for the number 13601.

Given Data / Assumptions:

• Dividend under consideration: 13601.
• Divisor: 87.
• We seek the least non-negative integer k such that (13601 - k) is divisible by 87.

Concept / Approach:By the remainder theorem of division: if N = 87 * q + r with 0 ≤ r < 87, then N - r = 87 * q is exactly divisible by 87. Therefore the minimal subtraction is simply the remainder r when 13601 is divided by 87. No larger subtraction is needed because r is, by definition, the smallest non-negative offset that lands on a multiple of 87.

Step-by-Step Solution:Divide 13601 by 87 to obtain the remainder r.Compute r = 13601 mod 87.The computation yields r = 29.Therefore, subtract k = 29: 13601 - 29 = 13572, which is divisible by 87.

Verification / Alternative check:Check divisibility directly: 13572 / 87 = 156 exactly, so the subtraction 29 is correct and minimal. Any smaller positive subtraction would leave a non-zero remainder.

Why Other Options Are Wrong:

• 49 / 23 / 31 / 58: These do not equal the true remainder 29. Subtracting any of these does not yield an exact multiple of 87 or is not the least such subtraction.

Common Pitfalls:Subtracting the next multiple difference in the wrong direction; mixing up the idea of “least addition” versus “least subtraction”; performing unnecessary long division without focusing on the remainder.

Final Answer:29