The sum of three consecutive even integers is 78. What is the largest of these integers?
["The Sum of Three Consecutive Even Integers is 78: What Is the Largest?", "Mathematics often presents us with puzzles that are not only engaging but also beneficial for sharpening problem-solving skills. One such intriguing question is: What are the three consecutive even integers whose sum is 78, and what is the largest of these? This article will guide you step-by-step through solving the problem, revealing the integers and confirming the largest one with clarity and confidence.", "---", "### Understanding Consecutive Even Integers", "Consecutive even integers are numbers that follow one another in order, increasing by 2 each time. For example, 4, 6, 8 — these are three consecutive even integers. Generalizing this, we can represent three consecutive even integers as:", "- First integer: ( x )\n- Second integer: ( x + 2 )\n- Third integer: ( x + 4 )", "---", "### Setting Up the Equation", "According to the problem, the sum of these three integers equals 78:", "[\nx + (x + 2) + (x + 4) = 78\n]", "Simplify the left side:", "[\nx + x + 2 + x + 4 = 78\n]\n[\n3x + 6 = 78\n]", "---", "### Solving the Equation", "Subtract 6 from both sides:", "[\n3x = 72\n]", "Divide both sides by 3:", "[\nx = 24\n]", "Now, compute the three consecutive even integers:", "- First integer: 24\n- Second integer: 24 + 2 = 26\n- Third (largest) integer: 24 + 4 = 28", "---", "### Verifying the Solution", "Add the three integers to confirm:", "[\n24 + 26 + 28 = 78\n]", "The sum is correct, verifying that the numbers are accurate.", "---", "### Conclusion", "The three consecutive even integers that sum to 78 are 24, 26, and 28.", "The largest of these integers is 28.", "This simple yet elegant problem demonstrates how algebraic expressions convert word problems into solvable equations, reinforcing foundational math skills useful in everyday life and advanced studies.", "---", "Ready to tackle similar math challenges? Explore more on integer sequences, algebra, and problem-solving strategies to build confidence and clarity in mathematics."]
