Problem 155

A train stops at Waterloo Station at 6:25 a.m. and every $3$ minutes thereafter, and so a train
stops only at times which differ from 6:25 a.m. by a multiple of $3$. Of the given answers, a train stops at
the station at 6:28 a.m. (difference is $3$ minutes), at 6:40 a.m. (difference is
$15$ minutes), and at 6:55
a.m. (difference is $30$
minutes).

A bus stops at Waterloo Station at 6:25 a.m. and every $5$ minutes thereafter, and so a bus stops
only at times which differ from 6:25 a.m. by a multiple of $5$. Of the given answers, a bus stops at
the station at 6:30 a.m. (difference is $5$ minutes), at 6:40 a.m. (difference is
$15$ minutes), and at 6:55 a.m.
(difference is $30$ minutes).

Thus, the next time that both a train and a bus stop at Waterloo Station
at the same time is 6:40 a.m.

Note: The lowest common multiple of both $3$ and $5$ is $15$. This tells us that a train and a bus
will stop at the station every $15$
minutes after 6:25 a.m. This confirms that 6:40 a.m. is the next time
that both a train and a bus will stop at the station at the same
time.