The sum of three consecutive even integers is 84. What is the product of the smallest and largest of these integers?
["Understanding the Sum of Three Consecutive Even Integers: Finding Their Product", "Mathematics often reveals elegant patterns through simple word problems. One classic example is finding three consecutive even integers whose sum equals 84—and discovering the product of the smallest and largest of these integers. This problem not only strengthens arithmetic reasoning but also illustrates how algebraic thinking simplifies everyday math challenges.", "In this article, we’ll explore how to determine the three consecutive even integers that add up to 84, then calculate the product of the smallest and largest numbers among them.", "---", "### Defining Consecutive Even Integers", "Consecutive even integers are numbers like 2, 4, 6 or 10, 12, 14—each differing by exactly 2. Mathematically, if we let the first even integer be ( x ), then the next two consecutive even integers are ( x + 2 ) and ( x + 4 ).", "They are consecutive because they follow each other in order without gaps, and “even” ensures all numbers are divisible by 2.", "---", "### Setting Up the Equation", "Given that the sum of these three integers is 84, we write:", "[\nx + (x + 2) + (x + 4) = 84\n]", "Simplify the left-hand side:", "[\nx + x + 2 + x + 4 = 84\n]\n[\n3x + 6 = 84\n]", "Subtract 6 from both sides:", "[\n3x = 78\n]", "Divide both sides by 3:", "[\nx = 26\n]", "So, the three consecutive even integers are:", "- First: ( 26 )\n- Second: ( 28 )\n- Third (largest): ( 30 )", "---", "### Calculating the Product of the Smallest and Largest", "Now that we know the integers are 26, 28, and 30, we compute the product of the smallest (26) and largest (30):", "[\n26 \ imes 30 = 780\n]", "This result follows from multiplying directly:", "[\n26 \ imes 30 = (25 + 1) \ imes 30 = 25 \ imes 30 + 1 \ imes 30 = 750 + 30 = 780\n]", "---", "### Why This Problem Matters", "Understanding how to express real-world conditions as equations is valuable in algebra and problem-solving. This particular example demonstrates:", "- How to model sequences of numbers with variables\n- How to use arithmetic to solve for unknowns\n- How properties of even numbers simplify calculations", "Moreover, the pattern of consecutive even integers appears in sports statistics, row numbering systems, and spacing problems—making this a practical and repeatable skill.", "---", "### Final Answer", "The sum of three consecutive even integers is 84 when the integers are 26, 28, and 30. The product of the smallest and largest is:", "[\n26 \ imes 30 = 780\n]", "This simple yet insightful problem shows how foundational math concepts help unlock clear, efficient solutions.", "---", "Keywords for SEO:\nthree consecutive even integers sum to 84, product of smallest and largest even integers, consecutive even integers math problem, word problem solving, integer sequences, algebra basics"]
