# Finding the solution for day k

There are 25 lights $$\{l_1, l_2, \dots , l_{25}\}$$ in off mode. On Day 1, every light is on. On the Day 2, every 2nd light. i.e, $$l_2, l_4, \dots , l_{24}$$, are off. On the 3rd day, every 3rd, i.e., $$l_3, l_6, \dots , l_{18}$$, are toggled (switched on if it is off/switched off if it is on). This process continues for 25 passes, such that on $$i$$th day, every $$i$$th light is toggled. Does there exist a day $$k$$ when all the lights will be in either on or off mode? How to explain this?

For few days, I am unable to think anything for this question. Could anyone tell me what can be done here?

• How about programming it and seeing what happens? You could even just work it out by hand, since the numbers are quite small. Jun 14 at 7:55
• Can you give the source of this problem ? Basically, you should be able to solve trivially this problem reasonning only on lights $l_1$ and $l_2$. Jun 14 at 7:56
• i found this question on discord given long back, after few iteration, there won't be any k day where either of them are on or off, but I am unable to understand, how to give the reason Jun 14 at 8:04
• just list all days of $l_1$ switch. Then list all days of $l_2$ switch. What can you conclude ? Jun 14 at 8:26