In a clock, a large gear completes a rotation every 45 seconds and a small gear completes a rotation every 18 seconds. How many seconds pass before the gears align again?

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Essentially you want to know what the least common multiple of their respective rate times ...

18=2*3*3 and 45=3*3*5 so the least common multiple is 2*3*3*5=90

check

90/45=2 rev

90/18=5 rev

Notice that these are complete revolutions which is what we want, complete revolutions at the same time ...