The smallest number of full rotations of the **24-tooth gear** required for alignment (so that both complete an integer number of cycles) is $oxed{3}$.

Tue, 05 May 2026 08:48:03 +0000

# The Smallest Number of Full Rotations of a 24-Toothed Gear for Perfect Alignment Achieving precise mechanical alignment in gear systems often depends on cyclic motion repetition. Consider a system involving a fixed and rotating 24-tooth gear engaged with another component—such as a slot, groove, or mesh mechanism. To ensure full synchronization where both gears complete an exact integer number of rotations, finding the smallest common multiple is essential. For a 24-tooth gear, each full rotation advances the gear by one full tooth cycle. The mechanical alignment occurs when both the gear and its engaged part complete whole rotations simultaneously. Since 24-toothed gears repeat alignment when their rotation counts share a common cycle, we compute the least common multiple (LCM) of the gear’s tooth count to find the minimal rotation threshold. The critical insight is that the smallest number of full rotations for the 24-tooth gear to realign with itself (and thus enable perfect synchronization in fixed partnerships) is defined by the LCM of 24 and 1—not the gear alone, but the joint periodicity required by the system. However, when aligning with a fixed external mechanism such as a rhythm or another gear phase, scalability via multiples applies. For full structural or cyclic synchronization where both components match their rotation cycles exactly, the minimal rotations reaching a shared integer baseline is exactly 3 full turns.