What is the least number that must be added to 8961 so that the result is exactly divisible by 84?

Introduction / Context:
This question is another example of using remainders and divisibility. You are asked to find the smallest number that must be added to a given integer so that the sum becomes divisible by a specified divisor. Problems like this train your ability to work with modular arithmetic and understand how to move from one multiple of a number to the next.

Given Data / Assumptions:
- The starting number is 8961.
- The divisor is 84.
- We need the smallest non negative integer k such that 8961 + k is exactly divisible by 84.
- This means (8961 + k) mod 84 must be 0.

Concept / Approach:
When a number N is divided by d, it can be written as N = d * q + r, where q is the quotient and r is the remainder. To reach the next multiple of d, we must add (d - r) if r is not zero. This will make the new remainder zero. Thus, the least number to be added is directly related to the remainder when the original number is divided by the divisor.

Step-by-Step Solution:
Step 1: Divide 8961 by 84 to find the remainder.Step 2: Compute 84 * 100 = 8400. The difference 8961 - 8400 = 561.Step 3: Now divide 561 by 84. We have 84 * 6 = 504, and 84 * 7 = 588 which is larger than 561, so 6 is the correct additional multiple.Step 4: The remainder at this stage is 561 - 504 = 57.Step 5: Therefore, 8961 = 84 * 106 + 57 (since 100 + 6 = 106).Step 6: To reach the next multiple of 84, the sum must be 84 * 107.Step 7: The difference between the next multiple and 8961 is 84 - 57 = 27.Step 8: Hence, the least number that must be added to 8961 is 27.

Verification / Alternative check:
After adding 27, the new number is 8961 + 27 = 8988. Check divisibility: 84 * 107 = 8988. Since this multiplication is exact with no remainder, 8988 is perfectly divisible by 84 and our added amount of 27 is correct. Any smaller number would leave a positive remainder, meaning the result would not be exactly divisible by 84.

Why Other Options Are Wrong:
Adding 57 would give 9018, which is 84 * 107 plus 30, so it is not the least number needed. Adding 141 or 107 jumps far beyond the nearest multiple and is unnecessary. Adding 21 results in 8982, which is 6 short of 8988 and not divisible by 84. These options either do not produce a multiple of 84 or do not represent the minimum required addition.

Common Pitfalls:
A common error is to miscalculate the remainder, especially when dealing with several steps of subtraction. Some learners try to guess the number to be added instead of following the remainder approach systematically. It is always safer to compute the exact remainder and subtract it from the divisor to find the minimal additional amount required.

Final Answer:
The least number that must be added is 27.