Understanding the Problem

* We have two gears rotating at different speeds.

* We need to find the least common multiple (LCM) of their rotation times to determine when they'll align again.

Calculations

1. Find the LCM: The LCM of 45 and 18 is 90.

2. Interpret the Result: This means that the larger gear will complete 2 rotations (90 seconds / 45 seconds per rotation = 2 rotations) and the smaller gear will complete 5 rotations (90 seconds / 18 seconds per rotation = 5 rotations) in 90 seconds.

Answer: The gears will align again after 90 seconds.