At the bus stop near his house, Quinzinho can catch buses from two lines to go to school. The buses of one line pass every 15 minutes and those of the other every 25 minutes, and at 7:30 AM the buses of both lines pass together.

(a) At what time will they pass together again?

(b) Between 7:30 AM and midnight, what are the times when the buses pass together at this stop near Quinzinho's house?
(a) Factoring, we have $15=3 \cdot 5$ and $25=5^{2}$, so the least common multiple of $15$ and $25$ is $75=3 \cdot 5^{2}$. Thus, the two buses will pass together at the point every 75 minutes, or 1 hour and 15 minutes. Therefore, the buses will pass together again near Quinzinho's house at $7 \mathrm{~h} 30 \mathrm{~min}+1 \mathrm{~h} 15 \mathrm{~min}=8 \mathrm{~h} 45 \mathrm{~min}$.

(b) To obtain the times when the buses will pass together at the bus stop near Quinzinho's house, we need to keep adding 1 hour and 15 minutes, resulting in 8:45 AM, 10:00 AM, 11:15 AM, 12:30 PM, 1:45 PM, 3:00 PM, 4:15 PM, 5:30 PM, 6:45 PM, 8:00 PM, 9:15 PM, 10:30 PM, and 11:45 PM. The next bus only passes after midnight.