Limousines travelling from the downtown hotel to the airport follow different routes. Those servicing Route A leave every 60 minutes, Route B every 30 minutes and Route C every 45 minutes. If the first departure on each route is 6:30am, when will be the next time limousines depart at the same time on all three route?

Suppose it happens x minutes after. Then x has to be divisible by 30, 45 and 60. Since we are looking for the next time it happens x has to be the smallest possible. So we need to find a least common multiple of 30, 45 and 60.
30 = 15 × 2
45 = 15 × 3
60 = 15 × 4
Therefore least common multiple is 15 x 3 x 4 = 180.
So in 180 minutes (3 hours) they will depart at the same time. 3 hours after 6:30 a.m. is 9:30 am.

The answer is 9:30 a.m.

30 = 15 × 2
45 = 15 × 3
60 = 15 × 4
Therefore least common multiple is 15 x 3 x 4 = 180.

Shouldn't he LCM be 15 x 2 x 3 x 4 = 360 and so the answer should be in 360 minutes (6 hours) they will depart at the same time. 6 hours after 6:30 a.m. is 12:30 pm.

The answer is 12:30 pm.

No.
180 / 30 = 6
180 / 45 = 4
180 / 60 = 3

Therefore 180 is a multiple of 30, 45, and 60. Besides, 180 < 360. So 360 is definitely NOT the LEAST common multiple.

30 = 2 × 3 × 5
45 = 3 × 3 × 5
60 = 2 × 2 × 3 × 5

The least common multiple is 3 × (2 × 2 × 3 × 5) = 3 × 60 = 180