Question: Which of the following day(s) can't be the last day of a century? Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday Justify your answer.
Answer:
The last day of a century can not be Tuesday, Thursday or Saturday.
A normal year has 365 days whereas a leap year has 366 days. Every year which is divisible by 4 is called a leap year. Also, every 4th century is a leap year but no other century is a leap year.
1 normal year = 365 days = 52 weeks + 1 day
1 leap year = 366 days = 52 weeks + 2 day
Thus, a normal year has 1 odd day whereas a leap year has 2 odd days.
100 years
= 76 normal years + 24 leap years
= 76*[52 weeks + 1 day] + 24*[52 weeks + 2 day]
= (76*52) weeks + 76 days + (24*52) weeks + 48 days
= 5200 weeks + 124 days
= 5217 weeks + 5 days
i.e. 100 years contain 5 odd days
Similarly,
200 years contain 10 odd days i.e. 3 odd days.
300 years contain 15 odd days i.e. 1 odd days.
400 years contain (20+1) odd days i.e. 0 odd days.
Note that 400 years contain one more leap year.
Also, we have Sunday for 0 odd day, Monday for 1 odd day, Tuesday for 2 odd days, and so on...
Thus, last day of first century is Friday. (5 odd days)
Last day of second century is Wednesday. (3 odd days)
Last day of third century is Monday. (1 odd days)
Last day of forth century is Sunday. (0 odd days)
Since the order is repeating in successive cycles, the last day of a century can not be
Tuesday, Thursday or Saturday.