Correct approach: The gear with 48 rotations/min makes a rotation every $ \frac{1}{48} $ minutes. The other every $ \frac{1}{72} $ minutes. They align when both complete integer numbers of rotations and the total time is the same. So $ t $ must satisfy $ t = 48 a = 72 b $ for integers $ a, b $. So $ t = \mathrm{LCM}(48, 72) $.

["Correct Approach: Harnessing Gear Ratios to Find Alignment in Rotations", "When analyzing mechanical systems involving gears, one crucial question often arises: When will two interlocking gears realign after starting simultaneously? Understanding this involves interpreting their rotation cycles mathematically and identifying the shortest time when both complete an exact number of rotations—this moment is the least common multiple (LCM) of their rotation periods.", "---", "### The Basics: Gear Rotations and Time Cycles", "Consider two gears with different rotation speeds:", "- Gear A completes 48 rotations per minute, meaning each rotation takes $ \frac{1}{48} $ minutes.\n- Gear B completes 72 rotations per minute, with each rotation taking $ \frac{1}{72} $ minutes.", "If both start synchronized at time $ t = 0 $, we want to find the first moment when both gears complete whole rotations at the same time—not just a fraction of a turn.", "---", "### Modeling Rotation Times Mathematically", "The elapsed time $ t $ (in minutes) when Gear A has completed $ a $ full rotations satisfies:\n$$\nt = 48a \quad \ ext{(since 48 rotations/minute)}\n$$\nSimilarly, for Gear B:\n$$\nt = 72b \quad \ ext{(72 rotations/minute)}\n$$", "For both gears to align at exact rotations, $ t $ must be a common multiple of $ 48a $ and $ 72b $, but more precisely, $ t $ must equal $ 48a = 72b $ for some positive integers $ a $ and $ b $. This means $ t $ is the least common multiple (LCM) of 48 and 72:", "$$\nt = \mathrm{LCM}(48, 72)\n$$", "---", "### Why the LCM?", "The LCM of two numbers is the smallest positive number divisible by both. In this context:", "- $ \mathrm{LCM}(48, 72) $ gives the shortest time $ t $ when Gear A has turned $ \frac{t}{1/48} = 48t $ full turns,\n- and Gear B has turned $ 72t $ full turns,\n- ensuring both rotations are whole numbers.", "This ensures their rotation cycles realign perfectly.", "---", "### Calculating the LCM", "To compute $ \mathrm{LCM}(48, 72) $, use prime factorization:", "- $ 48 = 2^4 \ imes 3^1 $\n- $ 72 = 2^3 \ imes 3^2 $\n- $ \mathrm{LCM} = 2^{\max(4,3)} \ imes 3^{\max(1,2)} = 2^4 \ imes 3^2 = 16 \ imes 9 = 144 $", "So, $ t = 144 $ minutes is the first moment both gears complete an integer number of rotations simultaneously.", "At $ t = 144 $:\n- Gear A: $ 48 \ imes 144 = 6912 $ rotations\n- Gear B: $ 72 \ imes 144 = 10368 $ rotations", "---", "### Practical Applications", "This concept applies widely:\n- Mechanical engineering: Gear synchronization in machinery\n- Robotics: Coordinating motion sequences\n- Timing systems: Aligning digital or analog pulses\n- Education: Teaching multiples, LCM, and periodicity", "Understanding the alignment via LCM helps design reliable timing systems and optimize mechanical efficiency.", "---", "### Summary", "To determine when two gears—running at 48 and 72 rotations per minute—align perfectly:", "- Find the smallest $ t $ such that $ t = 48a = 72b $ for integers $ a, b $.\n- This is $ t = \mathrm{LCM}(48, 72) = 144 $ minutes.\n- At $ t = 144 $, both gears complete whole rotations simultaneously, ensuring perfect alignment.", "Mastering this approach not only solves the mechanical problem but strengthens foundational math skills in periodicity and number theory.", "---", "Keywords: gear alignment, LCM of 48 and 72, rotation cycles, mechanical gears, periodic motion, synchronization time, math in engineering", "Meta Description: Find the correct approach to align gear rotations by computing the least common multiple of 48 and 72, revealing the shortest time when full rotations synchronize."]