Thus, after $ \boxed{144} $ seconds, both gears complete an integer number of rotations (48×3 = 144, 72×2 = 144) and align again. But the question asks "after how many minutes?" So $ 144 / 60 = 2.4 $ minutes. But let's reframe: The time until alignment is the least $ t $ such that $ 48t $ and $ 72t $ are both multiples of 1 rotation — but since they rotate continuously, alignment occurs when the angular displacement is a common multiple of $ 360^\circ $. Angular speed: 48 rpm → $ 48 \times 360^\

["Discover When Two Interlocking Gears Align Again: The Mathematics Behind Rotation Synchronization", "When two gears rotate, their alignment — returning to the exact same angular position relative to their start — depends on how their rotation cycles repeat. Suppose Gear A rotates at 48 revolutions per minute, and Gear B at 72 rpm. After $ t $ minutes, Gear A completes $ 48t $ rotations, and Gear B completes $ 72t $. They align again when both have completed an integer number of full rotations — not just any multiple, but the moment when their progress is synchronized in whole cycles.", "But more precisely, since both start aligned and rotate continuously, they realign precisely when both have advanced by an integer number of rotations — but more fundamentally, when their angular displacements match again modulo $ 360^\circ $. The key insight: angular speed determines how fast rotation progresses. Gear A turns $ 48 \ imes 360^\circ = 17,280^\circ $ per minute. Gear B turns $ 72 \ imes 360^\circ = 25,920^\circ $ per minute. The relative angular rate is $ 25,920 - 17,280 = 8,640^\circ $ per minute. But for alignment — meaning the same relative angular position — actually, they realign every time both have completed full rotations, because each full rotation brings them back to starting orientation.", "Wait — yes: after $ t $ minutes, Gear A has rotated $ 48t $ times, Gear B $ 72t $ times. They return to the same position $ \bmod 360^\circ $ whenever $ 48t $ and $ 72t $ are integers — but actually, even more: they realign geometrically whenever $ 48t $ and $ 72t $ are integers and the angular offset is zero modulo $ 360^\circ $, which happens exactly when $ 48t $ and $ 72t $ are both integers — but since $ t $ is continuous, this happens periodically.", "But the fundamental period — the smallest $ t > 0 $ such that both have completed an integer number of full rotations — is the least $ t $ such that $ 48t \in \mathbb{Z} $ and $ 72t \in \mathbb{Z} $? No — that’s not correct. $ t $ doesn’t need $ 48t $ integer unless we define “complete rotation” as termination. But they rotate continuously. So “complete an integer number of rotations” means $ 48t = m $, $ 72t = n $ for integers $ m,n $. So $ t = \frac{m}{48} = \frac{n}{72} $. So $ \frac{m}{48} = \frac{n}{72} \Rightarrow 72m = 48n \Rightarrow 3m = 2n $. Smallest positive integers: $ m=2, n=3 $, so $ t = \frac{2}{48} = \frac{1}{24} $ minutes.", "Thus, the two gears realign every $ \frac{1}{24} $ minutes — or 15 seconds — after they’ve each rotated an integer number of times. This is the least common multiple of their rotation cycle lengths in minutes.", "The rotation periods are $ \frac{1}{48} $ and $ \frac{1}{72} $ minutes.\nLCM of $ \frac{1}{48} $ and $ \frac{1}{72} $ is $ \frac{1}{\mathrm{GCD}(48,72)} $.\nNow, $ \mathrm{GCD}(48,72) = 24 $, so $ \mathrm{LCM}\left(\frac{1}{48}, \frac{1}{72}\right) = \frac{1}{24} $ minutes.", "But the question asks: “after how many minutes?” So the answer is $ \frac{1}{24} $ minutes.", "However, let’s reframe for clarity and impact: despite fractional simplicity, this reflects harmonic synchronization in mechanical systems. The alignment condition — both gears returning to identical orientation — occurs precisely at $ t = \frac{1}{24} $ minutes, as this is the smallest $ t $ such that both have completed whole rotations.", "To generalize: time until alignment = $ \frac{1}{\mathrm{GCD}(a,b)} $ minutes, where $ a $ and $ b $ are rotation rates in rpm, provided $ a,b $ integers.", "But wait — is this always true? Consider $ a=1 $, $ b=2 $. Then $ \mathrm{GCD}(1,2)=1 $, $ t=1 $ min. After 1 min: Gear A: 1 rotation, Gear B: 2 rotations — alignment at start — correct. If $ a=2 $, $ b=3 $, then $ t=2/2=1 $ min: A: 2 rotations, B: 3 — align. Correct. So yes, the time is $ \frac{1}{\mathrm{GCD}(a,b)} $ minutes.", "Here, $ a=48 $, $ b=72 $, $ \mathrm{GCD} = 24 $, so:", "[\n\ ext{Time until alignment} = \frac{1}{24} \ ext{ minutes}\n]", "This can be converted to seconds: $ \frac{1}{24} \ imes 60 = 2.5 $ seconds — short, but correct for high rotation speeds.", "Alternatively, some derive alignment from when angular positions match: $ 48t \equiv 72t \pmod{360} $, so $ 24t \equiv 0 \pmod{360} $? No — $ 48t - 72t = -24t $, and for same position: $ 48t \equiv 72t \pmod{360} \Rightarrow 24t \equiv 0 \pmod{360} \Rightarrow 24t = 360k \Rightarrow t = 15k $. Smallest $ t=15 $ seconds = $ \frac{1}{24} $ min — matches.", "Thus, the time until alignment is $ \frac{1}{24} $ minutes.", "Final Answer:\nAfter $ \frac{1}{24} $ minutes — or 2.5 seconds — both gears complete an integer number of rotations and realign.", "> Answer: $ \boxed{\frac{1}{24}} $ minutes", "---", "Enhanced Understanding:\nThis result illustrates the beauty of number theory in everyday mechanics. Despite high rotation speeds, synchronization follows elegant arithmetic principles. For interlocking systems, alignment depends not on speed magnitude but on the ratio of rotation rates — captured perfectly by the GCD.", "---", "Keywords: gear alignment, rotational symmetry, least common multiple, number theory in physics, $ \mathrm{GCD}(48,72) $, time between rotational alignments, continuous rotation synchronization.", "Note: For optimal clarity, some educational resources express time in seconds: $ 2.5 $ seconds, but the exact fractional form $ \frac{1}{24} $ min is mathematically precise.", "---", "Refined Realization:\nHowever, in engineering contexts, especially with multiple rotating components, alignment period is often expressed in seconds. But since the problem asks for “after how many minutes,” and the mathematical structure lies in rational rotation cycles, the precise answer remains $ \frac{1}{24} $ minutes.", "Thus, the final, exact answer is:", "[\n\boxed{\frac{1}{24}}\n]"]