The light of a maritime lighthouse flashes every 180 seconds. Its green light flashes every 56 seconds. At midnight, both lights go off at the same time. Can they trigger again after 360s? Justify
No, the lights of the maritime lighthouse cannot trigger again after 360 seconds.
The green light of the lighthouse flashes every 56 seconds, which means it will flash after 56, 112, 168, 224, 280, and 336 seconds. However, none of these intervals are divisible by 180 seconds, the interval at which the main light flashes.
The intervals at which the main light flashes are 180, 360, 540, 720, etc. seconds. So, the next time the main light will flash after midnight is at 360 seconds.
Since the smallest common multiple of 56 and 180 is 5040 seconds, we need to find the next time after 360 seconds when both lights will flash together, which is the next multiple of 5040.
360 + 5040 = 5400 seconds
5400 seconds is the next time when both lights will go off at the same time. Therefore, after 360 seconds, the lights of the maritime lighthouse will not trigger again together.
To determine if both lights can trigger again after 360 seconds, we need to find the least common multiple (LCM) of their respective intervals.
The light of the maritime lighthouse flashes every 180 seconds, while the green light flashes every 56 seconds. To find the LCM, we can list multiples of each interval until we find a common multiple.
Multiples of 180: 180, 360, 540, 720, 900, 1080, ...
Multiples of 56: 56, 112, 168, 224, 280, 336, 392, 448, 504, 560, ...
From the list, we can see that the first number that appears in both lists is 1,008 seconds. This means that both lights will flash again at the same time after 1,008 seconds.
Since 1,008 seconds is greater than 360 seconds, both lights cannot trigger again simultaneously within a 360-second interval.