Catching the bus through NT

Question

I found the following question (among others) in some our elementary number theory text:

On Monday, Tom caught the bus at 9:00 am. The bus comes to the station every 7 hours. When will Tom be able to catch the bus again at 9:00, for the first time after Monday?

I think the problem is supposed to be solved using properties of divisibility or do some modelling through some diophantine equation, but i do not know where to start. Could someone help ? thanks in advance!

Answer 1

Since the bus passes every $7$ hours, there will be an integer number of $7$-h intervals between the two catches. So the number of hours between the two catches will be a multiple of $7$.

On the other hand, since Tom took the bus at $9:00$ am and he is gonna catch it again at $9$ a.m., there will be an integer number of days (i.e. $24$-h periods) between the two catches. So the number of hours between the two catches will be a multiple of $24$

And since we need the first time we'll catch the bus again at $9:00$ am, we need the least common multiple of $7$ and $24$, i.e.: $$ lcm(7,24)=7\cdot24=168 \ hours $$ whish is $7$ days after Monday's first catch or $24$ buses after that.

Answer 2

HINT: For the train to be at the station at 9:00am again, it must be 24 hours, 48 hours, 72 hours, etc after the first one.

So just find a common multiple of 7 and 24 and you will find you answer