My mother and I has been trying or best to figure this out.
Light A, B, and C stays off for 1s then turns on.
When Light A is on it stays on for 1s. When Light B is on it stays on for 2s. When Light C is on it stays on for 4s.
If light A and B is off and light C is on for it's 4th second, how long till all of the lights will be on at the same time?
Our working:
0 = off 1 = on
S:$0\, 1\, 2\, 3\, 4\, 5$
A: $0\, 1 0\, 1 \,0 \,1$
B: $0\, 1\, 1\, 0 \,1\, 1$
C: $1\, 0\, 1\, 1 \,1\, 1$
The answer is $5$ seconds, we would like to know how to find it through calculation(maybe some sort of formula or equation?).
It's much easier to solve this problem if we ignore the first second of the story. We set $t=0$ as the moment when all the lights turn on. Also, time is quantized in this case, so we only look at integer values of time, and we look at the "beginning" of each second. So at $t=0$, all the lights are on.
At $t=1$, the first light is turned off. We see that the first light is on if and only if $t$ is even, which is to say that $t$ is divisible by two. Therefore, the next moment when all the light are on, has to be a number divisible by two. Similar logic can be used for the other lamps. Can you take it from here?