What is the least number that must be subtracted from 13601 so that the resulting number is exactly divisible by 87?

Introduction / Context:
This problem is about divisibility and remainders. We are given a large number and asked to find the smallest amount that must be subtracted so that the new number becomes exactly divisible by a given divisor. Such questions test understanding of the relationship between dividend, divisor, quotient and remainder.

Given Data / Assumptions:
Original number N = 13601.Required divisor = 87.We want N − k to be divisible by 87 for the smallest possible nonnegative integer k.

Concept / Approach:
When a number N is divided by a divisor d, we can write N = d * q + r, where q is the quotient and r is the remainder, with 0 ≤ r < d. If we want a number less than N that is divisible by d, the nearest multiple below N is N − r. Therefore, the least number to subtract from N to obtain a multiple of d is the remainder r itself. So the key step is to compute 13601 modulo 87.

Step-by-Step Solution:
Step 1: Divide 13601 by 87 to find the remainder.Step 2: Compute 87 * 150 = 13050 as a nearby multiple.Step 3: Subtract 13050 from 13601 to refine the remainder: 13601 − 13050 = 551.Step 4: Now divide 551 by 87: 87 * 6 = 522 and 87 * 7 = 609, which is too large.Step 5: So 551 = 87 * 6 + 29, and the remainder when 13601 is divided by 87 is 29.Step 6: Therefore, the largest multiple of 87 less than or equal to 13601 is 13601 − 29.Step 7: The least number to be subtracted from 13601 to make it divisible by 87 is k = 29.

Verification / Alternative check:
We can verify by computing (13601 − 29) / 87. The result is 13572 / 87. If we divide 13572 by 87, we get exactly 156 with no remainder. This confirms that 13572 is a multiple of 87 and that subtracting 29 is sufficient and correct. If we tried subtracting a smaller number such as 23 or 31, the resulting number would not be divisible by 87.

Why Other Options Are Wrong:
If we subtract 23, we get 13578; dividing this by 87 does not give an integer. Subtracting 31 gives 13570, which again is not a multiple of 87. Subtracting 37 gives 13564, which also fails the divisibility test. Only subtracting 29 leads to an exact multiple of the divisor.

Common Pitfalls:
A common mistake is to compute the quotient roughly and ignore the precise remainder. Some learners also try to work downward by trial and error from 13601, which is time consuming and error prone. Understanding the relation N = d * q + r and focusing on the remainder provides a fast and reliable path to the solution.

Final Answer:
The least number that must be subtracted from 13601 to make it divisible by 87 is 29.