What least number must be added to 213 so that the resulting sum is exactly divisible by 9?

Introduction / Context:
This question tests your understanding of the divisibility rule for 9 and your ability to adjust a number slightly so that it becomes a multiple of 9. You are asked to find the smallest number that must be added to 213 so that the sum is divisible by 9. Such problems are very common in aptitude tests and help reinforce patterns in digit sums and divisibility.

Given Data / Assumptions:

- The original number is 213. - We need to add a non negative integer k such that 213 + k is divisible by 9. - The smallest such k is required.

Concept / Approach:
The divisibility rule for 9 states that a number is divisible by 9 if and only if the sum of its digits is a multiple of 9. Therefore, we can find the sum of the digits of 213, determine the next higher multiple of 9 and then see how much needs to be added to the digit sum to reach that multiple. The same amount must be added to the original number to make it divisible by 9.

Step-by-Step Solution:
Step 1: Compute the sum of the digits of 213: 2 + 1 + 3. Step 2: 2 + 1 + 3 = 6. Step 3: The nearest multiple of 9 that is greater than or equal to 6 is 9. Step 4: To make the digit sum equal to 9, we must add 3 to the current digit sum of 6. Step 5: Adding 3 to the original number 213 gives 213 + 3 = 216. Step 6: Check the digit sum of 216: 2 + 1 + 6 = 9, which is a multiple of 9. Step 7: Therefore, 216 is divisible by 9, and 3 is the least number that must be added to 213.

Verification / Alternative check:
We can also verify by direct division. 216 ÷ 9 = 24, which is an integer, confirming that 216 is divisible by 9. If we tried adding 1 or 2 instead, we would get 214 or 215. The digit sums of 214 (2 + 1 + 4 = 7) and 215 (2 + 1 + 5 = 8) are not multiples of 9, so neither is divisible by 9. Thus 3 is indeed the smallest possible addition.

Why Other Options Are Wrong:
Adding 1 yields 214, which is not divisible by 9 since its digit sum is 7. Adding 2 gives 215, digit sum 8, also not divisible by 9. Adding 4 gives 217, with digit sum 10, again not a multiple of 9. Only adding 3 produces a digit sum that is a multiple of 9 and a number divisible by 9.

Common Pitfalls:
Some learners confuse the rules for divisibility by 3 and 9 or miscompute the digit sum. Others may attempt long division by 9 on several nearby numbers instead of using the quick digit sum method. Carefully applying the digit sum rule is faster and less error prone.

Final Answer:
The least number that must be added to 213 to make it divisible by 9 is 3, which corresponds to option A.