Three clocks were set to ring at intervals as follows:
the first after every 6 minutes
the second after every 15 minutes
the third after every 24 minutes
if the clocks were set at the same time, after how many minutes did them ring together?
What is the least common multiple of 6, 15, and 24?
To find the number of minutes after which all three clocks ring together, we need to find the least common multiple (LCM) of the intervals at which they ring.
The first clock rings every 6 minutes, so its multiples are 6, 12, 18, 24, 30, 36, ...
The second clock rings every 15 minutes, so its multiples are 15, 30, 45, 60, 75, ...
The third clock rings every 24 minutes, so its multiples are 24, 48, 72, 96, ...
To find the LCM, we need to find the smallest number that is a multiple of each of these intervals.
Starting with the smallest interval, 6 minutes, we can see that it is also a multiple of 6, so we keep it.
Moving to the next interval, 15 minutes, we find that it is not a multiple of 6, so we multiply it by 2 to get 30 minutes.
Moving to the next interval, 24 minutes, we find that it is not a multiple of 30, so we multiply it by 2 to get 48 minutes.
Continuing this process, we find that the next multiple of 48 is 96 minutes, which is also a multiple of 30.
Therefore, the LCM of 6, 15, and 24 is 96 minutes.
So, all three clocks will ring together after 96 minutes.