*Which can never be the last day(s) of the century?*

*You have to tell which of the seven day or days can never be the last of any century. You can consider 1 January 0001 to be a Monday for reference. After making a decent attempt yourself, scroll below for the answer.*

To answer this question we must first try to calculate how many days are there in a century.

In the most widely used calendar, the Gregorian calendar, there are two categories of years. One is an ordinary year which comprises of 365 days and another, a leap year which comprises of 366 days.

**What is the frequency of a leap year?
**A leap year occurs in all those years whose number is wholly divisible by 4. But there is an exception to this rule. The century years are not leap years unless they can be evenly divided by 400. For eg. the years, 4, 48, 400 are leap years and 5, 51, 100, 200 NOT.

**Let us calculate the number of days in a century now :
**In a period of 100(say 1 January 0001 – 31 December 0100

**– I**) years, there will be 24(all multiples of 4, except 100) leap years and 76 ordinary years. Hence the number of days will be:

**24*366 + 76*365 = 36524 days.**Similarly, for the periods between 101-200(say 1 January 0101 – 31 December 0200

**– II**) and 201-300(say 1 January 0201 – 31 December 0300

**– III**) the number of days would be 36524

**.**For the last century in our quartet of consideration, the number of days would be one more, i.e.

**36525**owing to the last year’s nature of being the leap year (the 400th year! –

**IV**).

**So there are two types of years and two types of centuries? Fascinating!**

Let us now zoom into our problem statement now. According to Wolfram alpha 1 January, 0001 was a Monday. Now for this century, we can calculate the first day of the next century by dividing the number of days in the century by 7 and check for the offset(remainder). **36524 mod 7 =** **5**, which means that the first day of the next century be Monday + 5 i.e. Saturday, making **Friday **the last day of the first century we considered**.**

Repeating the same process we will get the last day of the next two centuries to be a **Wednesday** and a **Monday.** For the last century of the 400 year period we will have another day to look forward to, hence the shift now becomes a +6, resulting in **Sunday **as the last day!

Voila! Guess what? We reached where we started! **Friday -> Wednesday -> Monday -> Sunday. **As this four hundred year cycle will repeat over and over we can conclude that for the Gregorian calendar the last day of any century can never be a **Tuesday, Thursday or a Saturday!**

**BONUS (Why leap years?)
**The Earth luckily(or sadly depending on which God you support) takes 365.2422 days to complete one revolution around the Sun. It was for this 0.2422 days that a complex system of leap year had to be introduced. Often leap seconds are also added in order to keep our calendar in sync with Earth’s.

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