The sum of three consecutive even integers is 84. What is the smallest of these three even integers?

Introduction / Context:
This question is a classic example of working with consecutive even integers and forming a simple linear equation. It tests understanding of how to represent consecutive even numbers algebraically and how to use a given sum to find each number.

Given Data / Assumptions:

- We have three consecutive even integers.

- The sum of these three integers is 84.

- We must find the least (smallest) of these three integers.

Concept / Approach:
Consecutive even integers can be represented as x, x + 2, and x + 4, where x is an even integer. Their sum is given to be 84. So we can set up the equation x + (x + 2) + (x + 4) = 84, solve for x, and then interpret x as the smallest of the three integers.

Step-by-Step Solution:
Step 1: Represent the three consecutive even integers. Let the smallest even integer be x. Then the next two even integers are x + 2 and x + 4. Step 2: Use the given sum. x + (x + 2) + (x + 4) = 84. Step 3: Simplify the equation. Combine like terms: x + x + x = 3x and 2 + 4 = 6. So 3x + 6 = 84. Step 4: Solve for x. Subtract 6 from both sides: 3x = 84 - 6 = 78. Divide by 3: x = 78 / 3 = 26. Step 5: Identify the three integers. They are 26, 28, and 30. So the smallest integer is 26.

Verification / Alternative check:
Check the sum: 26 + 28 + 30 = 84. This matches the given condition exactly, so our result is consistent.

Why Other Options Are Wrong:
- 28: If the smallest were 28, the others would be 30 and 32, and the sum would be 90, not 84.
- 30: With 30, 32, and 34, the sum would be 96.
- 32: With 32, 34, and 36, the sum would be 102.

Common Pitfalls:
Some students mistakenly use x, x + 1, x + 2 for consecutive even integers, which is incorrect because that represents consecutive integers, not even ones. Others misread the question and give the middle or largest integer instead of the smallest. Keeping track of what is asked and representing the pattern correctly avoids these mistakes.

Final Answer:
The smallest of the three consecutive even integers is 26.