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The picture above shows a part of a mechanical contraption exhibited in the main gallery of the MIT museum. The contraption begins with a motor (on the left) turning at \(212\ \textrm{rpm}\). It is connected to a sequence of \(N\) worm-gears, which can be seen in the picture. Each worm-gear pair reduces the rotation speed by a factor of \(f=50\). The axle of the last gear in the sequence is embedded in a concrete wall that cannot rotate at all.

Find the smallest \(N\) that will suffice to keep the exhibit running for a century.


  • Each wheel has approximately 50 teeth.
  • The system jams when the final wheel needs to rotate by half a tooth.

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