What is the smallest number that must be subtracted from 210 so that the resulting number is exactly divisible by 11?

Introduction / Context:
This problem involves divisibility and remainders and asks how to adjust a number by subtraction so that the result becomes divisible by a given divisor. It reinforces modular arithmetic concepts used in many aptitude tests.

Given Data / Assumptions:
- Original number: 210.
- Divisor: 11.
- We must find the least non negative integer k such that 210 − k is divisible by 11.

Concept / Approach:
If a number N leaves a remainder r when divided by d, then subtracting r from N makes the result exactly divisible by d. Here we first find the remainder when 210 is divided by 11. That remainder will be the smallest number that needs to be subtracted to obtain a multiple of 11.

Step-by-Step Solution:
Let N = 210 and d = 11. Divide 210 by 11: 11 × 19 = 209. Compute the remainder: 210 − 209 = 1. Therefore N leaves a remainder r = 1 when divided by 11. Subtracting r gives 210 − 1 = 209, which is divisible by 11.

Verification / Alternative Check:
We quickly check: 209 ÷ 11 = 19 exactly, with no remainder, confirming that 209 is a multiple of 11. Any smaller non negative subtraction than 1 would be 0, and 210 itself is not divisible by 11. Any larger subtraction such as 2 or 3 would also give multiples of 11, but they would not be minimal because 1 already works.

Why Other Options Are Wrong:
Subtracting 2 gives 208, which is not divisible by 11 because 11 × 18 = 198 and 11 × 19 = 209.
Subtracting 3 produces 207, which is also not an exact multiple of 11.
Subtracting 4 yields 206, again leaving a remainder when divided by 11.
Subtracting 5 gives 205, which is divisible by 5 but not by 11. Only subtracting 1 gives the nearest lower multiple of 11.

Common Pitfalls:
Students sometimes mistakenly add the remainder rather than subtract it when the question explicitly asks for subtraction. Others may perform incorrect long division and find a wrong remainder. Carefully computing the product of the divisor and the integer quotient, then subtracting from the original number, helps avoid such calculation errors.

Final Answer:
The smallest number that must be subtracted from 210 to make it divisible by 11 is 1.