Prove that last day of a century cannot be Tuesday, Thursday or Saturd

Prove that last day of a century cannot be Tuesday, Thursday or Saturday.

Answer»

Sol. 100years have = 5 odd days ∴ Last days of Ist century is Friday

200years have = 10 odd days ∴ Last day of IIst century is Wednesday

= 3 odd days

300years have = 15 odd days ∴ Last day of IIIrd century is Monday

= 01 odd day

400years have = (5 × 4 + 1) Last day of 4th century is Sunday

= 21 odd days

= 0 odd days

Since the order keeps on cycling, we see that the lat day of the century cannot be Tuesday, Thursday or Saturday.

						Tables: For calculating odd days
Month	Jan	Feb	March	April	May	June	July	Aug	Sep	Oct	Nov	Dec
Odd days	3	0/1 ord./Leap yr	3	2	3	2	3	3	2	3	2	3

Months of years	Ist three months 1 Jan to 31 March	IInd Three months 1 Apr to 30 Jun	IIIrd three months 1 July to 30 Sep.	Ivth (last) three months	Total year 1 Jan to 31 Dec.
Total days	90 / 91 ord / leap	91	92	92	365/366 ord./leap
Odd days	6 / 0 ord./leap	0 odd day	1 odd day	1 odd day	1 / 2 odd days