Aptitude Topic - Calendars
Years can be classified into two types, a leap year and a non-leap year. A leap year has 366 days while a non-leap year has 365 days.
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For an year to be a leap year, it must be divisible by 4 but for centuries, it must also be
divisible by 400.
So the years 1700 and 2200 are not leap years as they are not divisible by 400.
Odd days
It is the number of days left after a week, i.e., the remainder we get when we divide a given number of days by 7. For example, in 72 days we have 2 odd days.
| Odd day | Associated day of the week |
|---|---|
| 0 | Sunday |
| 1 | Monday |
| 2 | Tuesday |
| 3 | Wednesday |
| 4 | Thursday |
| 5 | Friday |
| 6 | Saturday |
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A leap year has 2 odd days ( 366 % 7 ) while a non-leap year has 1 odd day.
| Number of years | Total number of odd days |
|---|---|
| 100 | 5 |
| 200 | 3 |
| 300 | 1 |
| 400 | 0 |
What was the day of the week on 21st June, 2019?
The steps we must follow are:
- Write the closest leap year which is a multiple of 400 as it will have 0 odd days
- Find the odd days for the rest of the years (taking into consideration the leap years) and the months.
- Check the day of the week corresponding to the final number of odd days we got.
21June2019 = 2000years + 18years + Jan + Feb + Mar + Apr + May + 21daysofJune
oddDays = 0 + 22 + 3 + 0 + 3 + 2 + 3 + 0 = 33
If we divide by 7 again, we get the final odd days left as
oddDays = 5
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Explanation
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2000 years have 0 odd days. Then, till 2019 , 18 years have gone by which contribute 22 odd
days (4 leap years + 14 non-leap years). January , March and May give 3 odd days ( 31 % 7 )
and April gives 2 odd days ( 30 % 7 ). As 2019 is non-leap, February gives 0 odd days
( 28 % 7). As for the 21 days in June, they too give 0 odd days. As the number of odd days
we got is 5, the 21st day of June will be a Friday.
Dooms Day method
This is a method to find the day of the week of a particular date.
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Last day of February = 4th day of April = 6th day of June = 8th day of August = 10th day of October = 12th day of December
You can remember this as:
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Last day of February = 4/4 = 6/6 = 8/8 = 10/10 =12/12
Where the numerator refers to the day and the denominator to the month.
The last day of February is called Dooms Day and can be calculated according to the equation given below:
Doomsday = Anchorday + (YY/12) + R(YY/12) + [R(YY/12)]/4
| Range of years | Number of Anchor days |
|---|---|
| 1800 - 1899 | 5 |
| 1900 - 1999 | 3 |
| 2000 - 2099 | 2 |
| 2100 - 2199 | 0 |
YY are the last two digits of the year for which we are finding and R is the reminder we get when we divide YY by 12.
What was the day of the week on 28th June, 2019?
Doomsday = 2 + 19/12 + [R.(19/12)] + [R.(19/12)]/4
Doomsday=2+1+7+1=11
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Here the year is 2019, so the Anchor day will be 2 and YY will be 19. The value of R will
be 19%12 which is equal to 7.
We take only the integer values from the division.
Now we got 11 days which has 4 odd days (11 % 7), the 4th odd day is a Thursday. Hence, the last day of February in 2019 was a Thursday. This implies that the 6th day of June is a Thursday, so is the 13th, 20th and 27th day of June. Thus June 28 will be a Friday.
Different type of question: In what future year can we reuse the calendar of 2007 ?
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Jan 1st 2007 --> Monday
2008 --> Tuesday
2009 --> Thursday (As 2008 is leap year and gives 2 odd days)
2010 --> Friday
2011 --> Saturday
2012 --> Sunday
2013 --> Tuesday (As 2012 is a leap year)
2014 --> Wednesday
2015 --> Thursday
2016 --> Friday
2017 --> Sunday (As 2016 is a leap year)
2018 --> Monday
So 2007 ‘s calendar can be reused in 2018. In questions such as above we can only match a leap year with another leap year and a non-leap year with another non-leap year, i.e, if we started with a leap year, we must end at a leap year and NOT at a non-leap which has same week day.