There are four bulbs A, B, C and D in a room. A glows for 2 minutes, then goes off for 1 minute and the process continues in this manner. Similarly, B glows for 3 minutes and then goes off for 1 minute and the process continues. For C, it is 4 minutes and 1 minute, while for D, it is 5 minutes and 1 minute respectively for glowing and going off. A man enters the room and comes out of it as soon as he completes an hour. What is the maximum number of times that he may face complete darkness in the room?
- A3
- B2
- C0
- D1
Explanation
A is off when t mod 3 = 2, B when t mod 4 = 3, C when t mod 5 = 4 and D when t mod 6 = 5. Combining t = 5 (mod 6) with t = 3 (mod 4) gives t = 11 (mod 12); with t = 4 (mod 5) this gives t = 59 (mod 60), the unique solution modulo 60. Within the hour [0, 60) minutes, all four bulbs are simultaneously off only around t = 59. Hence the man can face complete darkness at most once.