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The year consists of 365 days, 5 hours, 48 minutes (52 weeks and 1 odd day). An extra day is added once in every fourth year which was called the leap year, which has 366 days (52 weeks and 2 odd days).
To find the day of any given date of the year, you need to understand the calendar calculations:
1. First thing to remember, first January 1 AD was Monday therefore, we must count days from Sunday. This means the 0th day was Sunday, so the 7th day was Sunday again and so on and so forth.
2. The day gets repeated after every seventh day (concept of a week), if today is Monday, then 28th day from now will also be Monday as it a multiple of 7 (28/7 = 4, so four weeks). Here the 30 day will be calculated by 30/7, which is 4 weeks and 2 days, these two days are called odd days. With starting day as Monday and two odd days, the day will be Wednesday; this point is the most critical in calendars. The other of looking at it is since the 28th day is Monday, so the 30th day will be Wednesday. But you have to understand and use the concept of odd days as the question may be about thousands of years.
3. In a normal year there are 365 days so 52 weeks and 1 odd day, in a leap year there are 366 days so 52 weeks and 2 odd days.
4. In 100 years there are 24 leap years and 76 normal years, so the number of odd days are 24(2) + 76 = 124, which is 17 weeks + 5 odd days, so 100 years have 5 odd days.
5. In 200 years the number of odd days is twice the number in 100 years which is 10, which is one week and 3 odd days, so 200 years have 3 odd days. In 300 years, the number of odd days is 15, which is two weeks and 1 odd day, so 300 years have one odd day.
6. 400 year is a leap year; similarly the multiples of 400 are also leap years.
7. In 400 years, the number of odd days become 20 + 1(from the leap year), so total days are 21, which is three weeks and 0 odd days. In 400 years there are 0 odd days

Illustrations:

Q.1: Find the day of the week on 16th july, 1776.

Ans:
Tuesday
Explanation: 16th july, 1776 means = 1775 years + period from 1st january to 16th july. Now, 1600 years have 0 odd days.100 years have 5 odd days.75 years = 18 leap years + 57 ordinary years= (36 + 57) odd days = 93 odd days= 13 weeks + 2 odd days = 2 odd daysTherefore, 1775 years have (0 + 5 + 2) odd days = 0 odd days. Now, days from 1st Jan to 16th july; 1776Jan Feb March April May June July31 + 29 + 31 + 30 + 31 + 30 + 16 = 198 days= (28 weeks + 2 days) odd days. So total number of odd days = 2. Therefore, the day of the week was Tuesday.

Q.2: What was the day on 25th January, 1975?

Ans. Saturday

Explanation:
Counting the years 1600 + 300 + 74
In 1600 years, there are zero odd days
In 300 years, there is one odd day
In 74 years, there are 18 leap years and 56 normal years, so the odd days are:
18(2) + 56(1) = 36 + 56 = 92,
Which is 13 weeks and 1 odd day
In 25 days of January, 1975, there are 3 weeks and 4 odd days
Total odd days = 0 + 1 + 1 + 4, six odd days, so it was a Saturday.

Q.3: How many times does the 29th day of the month occur in 400 consecutive years?

Ans. 4497

Explanation:
In 400 consecutive years there are 97 leap years. Hence, in 400 consecutive years February has the 29th day 97 times and the remaining eleven months have the 29th day 400 × 11 or 4400 times.
Thus the 29th day of the month occurs = 4400 + 97 = 4497 times.

Q.4: Today is 3rd November. The day of the week is Monday. This is a leap year. What will be the day of the week on this date after 3 years?

Ans. Thursday

Explanation:This is a leap year. So, none of the next 3 years will be leap years. Each year will give one odd day so the day of the week will be 3 odd days beyond Monday i.e. it will be Thursday.

Q.5: December 9, 2001 is Sunday. What was the day on December 9, 1971?

Ans. Tuesday.

Explanation:
Total number of days = 30 × 365 + 8 days from leap years = 10958
Thus number of weeks = 1565. Hence December 9, 1971 must have been Tuesday.

Q.6: If today is Monday, what will be the day 350 days from now?

Ans. 350 days, 350/7 = 50, no odd days, so it will be a Monday.

Q.7:
If today is Monday, what will be the day one year and 50 days from now?

Ans. Cannot be determined, as we don’t know if it is a leap year or not.

Formula for years
Odd Days:

We are supposed to find the day of the week on a given date.
For this, we use the concept of 'odd days'.
In a given period, the number of days more than the complete weeks are called odd days.

Leap Year:

(i). Every year divisible by 4 is a leap year, if it is not a century.
(ii). Every 4th century is a leap year and no other century is a leap year.

Note: A leap year has 366 days.

Examples:

1. Each of the years 1948, 2004, 1676 etc. is a leap year.
2. Each of the years 400, 800, 1200, 1600, 2000 etc. is a leap year.
3. None of the years 2001, 2002, 2003, 2005, 1800, 2100 is a leap year.

Ordinary Year:

The year which is not a leap year is called an ordinary year. An ordinary year has 365 days.

Counting of Odd Days:

1 ordinary year = 365 days = (52 weeks + 1 day). Therefore 1 ordinary year has 1 odd day.

1 leap year = 366 days = (52 weeks + 2 days). Therefore 1 leap year has 2 odd days.

100 years = 76 ordinary years + 24 leap years
= (76 x 1 + 24 x 2) odd days = 124 odd days.
= (17 weeks + days) 5 odd days.

Therefore, Number of odd days in 100 years = 5
Similarly, Number of odd days in 200 years = (5 x 2) 3 odd days.
And, Number of odd days in 300 years = (5 x 3) 1 odd day.
Also, Number of odd days in 400 years = (5 x 4 + 1) 0 odd day.
Similarly, each one of 800 years, 1200 years, 1600 years, 2000 years etc. has 0 odd days.
Day of the Week Related to Odd Days:

 No. of days: 0 1 2 3 4 5 6 Day: Sun. Mon. Tues. Wed. Thurs. Fri. Sat.

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