On the coast there are three lighthouses.

The first light shines for 3 seconds, then is
off for 3 seconds.
The second light shines for 4 seconds, then is
off for 4 seconds.
The third light shines for 5 seconds, then is
off for 5 seconds.
All three lights have just come on together.
When is the first time that all three lights will be off? please help

LCM(3,4,5) = 60

To determine when all three lights will be off at the same time, we first need to find the least common multiple (LCM) of their on-off cycles.

The first light has an on-off cycle of 3 seconds. This means it repeats every 3 seconds.
The second light has an on-off cycle of 4 seconds. This means it repeats every 4 seconds.
The third light has an on-off cycle of 5 seconds. This means it repeats every 5 seconds.

To find the LCM of these three cycles, we can list the first few multiples of each cycle and find the smallest common multiple:

Multiples of 3: 3, 6, 9, 12, 15, ...
Multiples of 4: 4, 8, 12, 16, 20, ...
Multiples of 5: 5, 10, 15, 20, 25, ...

From the lists above, we can see that the smallest common multiple of 3, 4, and 5 is 60. Therefore, all three lights will be off at the same time after 60 seconds.

In conclusion, the first time that all three lights will be off together is after 60 seconds.

To determine when all three lights will be off at the same time, we need to find the least common multiple (LCM) of the time intervals in which each light is on/off.

The pattern for the first light is on for 3 seconds and off for 3 seconds.
The pattern for the second light is on for 4 seconds and off for 4 seconds.
The pattern for the third light is on for 5 seconds and off for 5 seconds.

To find the LCM, we need to find the smallest number that is divisible by 3, 4, and 5. Let's calculate it step by step:

1. Start by listing the multiples of each number:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27...
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36...
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45...

2. Look for the first number that appears in all three lists. In this case, it is 12.

Therefore, the first time all three lights will be off simultaneously is after 12 seconds.

Note: If you want to verify this, you can observe the pattern by counting the seconds for each light. After 12 seconds, all three lights will be off, and the pattern will repeat again.