Professor Shyam was talking to the students about 21st century which has started with a Monday. Incidently, he posed a question that the last day of the century cannot be
When number of days in a given period of time is divided by 7 then the remainder which results represents the number of odd days.
To start with consider a span of 100 years. Every 4th year is a leap year within a century and every 4th century year is a leap year.
This means years 4,8,12 etc are leap years while 100 is not a leap year.
But 400th year, 800th year etc are leap years.
By above argument 100 years contain 24 leap years and 76 non leap years. (Years 4,8,12....96 are leap years and 100th year is not. Therefore, number of leap years in 100 years is 100/4 - 1)
Number of days in 100 years = 24 x 366 + 76 x 365 = 36524
Dividing 36524 by 7, we will get a quotient of 5217 and remainder of 5.
Therefore, 100 years i.e., 36524 days has 5 odd days to end with. Since the century has started with monday, odd days in order will be Mon, Tue, Wed, Thu and Fri.
Last odd day will be the last day of the century. Hence, last day of 1st century will be Friday.
By similar ways one can find that 200 years will contain 3 odd days, 300 years will contain 1 odd day and 400 years will contain 0 odd days.
This means last day of 2nd century will be Wednesday, last day of 3rd century will be Monday and last day of 4th century will be Sunday.
The entire cycle will repeat for the next 400 years, thereafter next 400 years and so on.
Therefore, last day of the century cannot be Tuesday, Thursday or Saturday./p>