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Geometry Difficulty 3.6 AMC 10/12 Find the answer Canada

Two circular dials are positioned next to each other, and an
arrow is drawn on each dial, as shown.

The left dial rotates counterclockwise at $20\degree$ per second and the right dial
rotates clockwise at $8\degree$ per
second. What is the minimum number of seconds that must pass before the
arrows are pointing directly towards each other again?

Pick one

Solution

For the two arrows to point directly toward each other again,
they must each return to their original positions at exactly the same
time.

The left dial makes one complete rotation every $\dfrac{360\degree}{20\degree}=18$ seconds,
and so it returns to its original position at $18$ s, $36$ s, $54$ s, $72$ s, $90$ s, and so on. The right dial makes one
complete

rotation every $\dfrac{360\degree}{8\degree}=45$ seconds,
and so it returns to its original position at $45$ s, $90$ s, and so on.

Comparing the two lists of times, the minimum number of seconds that
must pass before the arrows are pointing directly toward each other
again is $90$ .

(Note that $90$ is the lowest common
multiple of $18$ and $45$ .)

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.