There are light bulbs from A to F and bulb A is always turned on.
Bulb B repeats "it is turned on for one minute and turned off for one minute.”
Bulb C repeats"it is turned on for one minute and turned off for two minutes."
Bulb D repeats "it is turned on for one minute and turned off for three minutes."
Bulb E repeats "it is turned on for one minute and turned off for four minutes.”
Bulb F repeats "it is turned on for one minute and turned off for five minutes."
All bulbs are turned on from 10:00 to 10:01.
(1) What time is it in the status which is all bulbs are on next time?
(2) What time is it in the status which is only A,C,E are on at the first time after 10:00?
(3) What time is it in the status which is only D is off at the first time after 10:00?
(4) How many minutes are there in the status which is only A is on from 10:00 to 11:00?
Bulb B repeats "it is turned on for one minute and turned off for one minute.”
Bulb C repeats"it is turned on for one minute and turned off for two minutes."
Bulb D repeats "it is turned on for one minute and turned off for three minutes."
Bulb E repeats "it is turned on for one minute and turned off for four minutes.”
Bulb F repeats "it is turned on for one minute and turned off for five minutes."
All bulbs are turned on from 10:00 to 10:01.
(1) What time is it in the status which is all bulbs are on next time?
(2) What time is it in the status which is only A,C,E are on at the first time after 10:00?
(3) What time is it in the status which is only D is off at the first time after 10:00?
(4) How many minutes are there in the status which is only A is on from 10:00 to 11:00?
Answer
(1) 11:00
(2) 10:15
(3) 10:30
(4) 16 minutes
(2) 10:15
(3) 10:30
(4) 16 minutes
Solution
(1) Confirm the cycle of the time between turning on and next turning on concerning bulb B ~ F respectively.
B : 1 + 1 = 2 minutes,、
B : 1 + 1 = 2 minutes,、
C : 1 + 2 = 3 minutes,
D : 1 + 3 = 4 minutes,
E : 1 + 4 = 5 minutes,
F : 1 + 5 = 6 minutes
The next time when all bulbs are on is to be the least common multiple of each cycle of turning on time which is 2, 3, 4, 5, 6.
Since the least common multiple is 60, next time is 10:00 + 60 minutes = 11:00.
The next time when all bulbs are on is to be the least common multiple of each cycle of turning on time which is 2, 3, 4, 5, 6.
Since the least common multiple is 60, next time is 10:00 + 60 minutes = 11:00.
(2) As bulb A is always on, examine the time while C and E is on and B, D and F is off.
The time cycle when C and D is on at the same time is the least common multiple of 3 and 5, which is 15.
The time cycle when C and D is on at the same time is the least common multiple of 3 and 5, which is 15.
Since 15 is not a multiple either of 2, 4, 6, it is found that B, D, and F are not on after 15 minutes.
The time is 10:00 + 15 minutes = 10:15.
(3) It is considered in the same way as (2), find the least common multiple of the time cycle of B, C, E, F and if it is not the multiple of 4, it is the time to be found.
The least common multiple of 2, 3, 5, 6 is 30 and 30 is not the multiple of 4 which means that after 30 minute D in not on.
The least common multiple of 2, 3, 5, 6 is 30 and 30 is not the multiple of 4 which means that after 30 minute D in not on.
The time is 10:00 + 30 = 10:30.
(4) In order to find the time while only A is on, we should find such time as any time cycle of turning on of B ~ F does not match from 10:00 until 11:00.
Since the time span to be examined is 60 minutes, in the numbers of 1 ~ 60 we should find the numbers which are not multiple of either of 2, 3, 4, 5, 6.
Since 4 and 6 are multiples of 2 or 3, we should count such numbers as not being the multiple of 2, 3 and 5.
The number of multiple of 2 is 30 (60 / 2 =30).
The number of multiple of 3 is 20 (60 / 3 = 20). The number of multiple of 5 is 12 (60 / 5 = 12).
The number of multiple of 6 which is the common multiple of 2 and 3 is 10 (60 / 6 = 10).
The number of multiple of 10 which is the common multiple of 2 and 5 is 6 (60 / 10 = 6).
The number of multiple of 15 which is the common multiple of 3 and 5 is 4 (60 / 15 = 4).
The number of multiple of 30 which is the common multiple of 2, 3 and 5 is 2 (60 / 30 = 2).
The total number of multiple of 2, 3 and 5 is (30 +20 + 12 ) - (10 + 6 ;4) + 2 = 44.
As 44 is the time while any bulb of B ~ F is on, the time while any bulbs other than A is off is 60 - 44 = 16 minutes.
