What is the least number that must be subtracted from 3401 so that the resulting number is exactly divisible by 11?

Introduction / Context:
This question checks your understanding of divisibility by 11 and your ability to adjust a given number so that it becomes a multiple of 11. You are asked for the least number that must be subtracted from 3401 in order to make the result divisible by 11. Such problems are common in aptitude exams and help reinforce modular arithmetic and divisibility rules.

Given Data / Assumptions:

- The original number is 3401. - We want to find a non negative integer k to subtract. - The number 3401 - k should be divisible by 11. - The value of k should be as small as possible.

Concept / Approach:
One straightforward approach is to find the remainder when 3401 is divided by 11. If a number N gives remainder r when divided by 11, then N - r is divisible by 11. Thus, the smallest non negative number that must be subtracted to make N divisible by 11 is exactly this remainder. We can compute the remainder through standard long division or by quick mental calculation using nearby multiples of 11.

Step-by-Step Solution:
Step 1: Divide 3401 by 11 to find the quotient and remainder. Step 2: Note that 11 * 300 = 3300. Subtracting from 3401 gives 3401 - 3300 = 101. Step 3: Now divide 101 by 11. Since 11 * 9 = 99, we have 101 - 99 = 2 as the remaining part. Step 4: Thus 3401 = 11 * 309 + 2. Step 5: The remainder when 3401 is divided by 11 is 2. Step 6: To make the number divisible by 11, we must subtract this remainder. Step 7: So, the required number to subtract is k = 2 and the resulting number is 3401 - 2 = 3399. Step 8: 3399 is divisible by 11 because 3399 / 11 = 309 exactly.

Verification / Alternative check:
As a second check, apply the divisibility rule for 11 to 3399. For 3399, sum of digits in odd positions (from left) is 3 + 9 = 12, and sum of digits in even positions is 3 + 9 = 12. The difference is 12 - 12 = 0, which is a multiple of 11, confirming that 3399 is divisible by 11. Doing the same for 3401 yields a non zero difference, so 3401 itself is not divisible by 11. This confirms that subtracting 2 is necessary and sufficient.

Why Other Options Are Wrong:
If we subtract 0, we stay at 3401, which is not divisible by 11. Subtracting 1 gives 3400, which upon division by 11 leaves a remainder. Subtracting 3 yields 3398, which also is not a multiple of 11. Only subtracting 2 brings the number to the nearest lower multiple of 11, namely 3399. Therefore the least required subtraction is 2.

Common Pitfalls:
Some learners may instead try to add numbers to 3401, or they may apply the divisibility rule incorrectly by miscounting positions of digits. Others might guess without checking the remainder carefully. Remember that checking the exact remainder is a quick and reliable method, and subtracting that remainder will always yield the nearest smaller multiple of the divisor.

Final Answer:
The least number that must be subtracted from 3401 is 2, which corresponds to option C.