There are 100 light bulbs lined up in a row. You start with all the lights off. You walk down the row and turn on every bulb. Then you go back to the beginning and switch off every second bulb. On your third pass, you toggle every third bulb (turning it off if it's on or turning it on if it's off). You continue this process for 100 passes. After completing the 100th pass, how many light bulbs are left on?
To solve this problem, let's think about factors:

1. A light bulb is toggled each time one of its factors is encountered during the passes.
2. For most numbers, their factors come in pairs (e.g., 2 and 3 are factors of 6, as well as 1 and 6).
3. However, perfect squares have an odd number of factors because one factor is repeated twice (e.g., 1, 3, and 9 are factors of 9, where 3 is repeated).
4. Since a light bulb starts in the off position, it will be on after an odd number of toggles and off after an even number of toggles.
5. Therefore, only the bulbs that are perfect squares will be left on after 100 passes because they have an odd number of factors.

Now let's count the perfect squares between 1 and 100:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

There are 10 perfect squares in this range.

So, after completing the 100th pass, there will be 10 light bulbs left on.