Here we will deal with finding the day of the week on a particular date. The process lies on obtaining the no. of odd days.

**Odd Days :- **In a given period, number of days more than a complete weeks are called odd days.

**Leap Year :-** Every year which is divisible by 4 is called a leap year. Thus 1992,1996,2004 etc. are leap years. Every 4^{th} century is a leap year like 400,800,1200 etc. but no other century is a leap year like 1500,1900,2100 etc.

An year which is not a leap year is called an **ordinary year**.

- An ordinary year has 365 days.
- A leap year has 366 days.

**Counting of odd days :**

1 ordinary year = 365 days

= 52 weeks + 1 day

= 1 odd day.

1 leap year = 366 days

= 52 weeks + 2 day

= 2 odd days.

100 years = 76 ordinary years + 24 leap years

= (76 x 52) + 76 days + (24 x 52) + 48 days

= 124 days = 124/7 = 5 odd days

**100 years = 5 odd days**

**200 years = (5+5)/7 = 3 odd days**

**300 years = (5+5+5)/7 = 1 odd day**

**400 years = 21 = 0 odd days**

**(7n+m) odd days, where m<7 is equivalent to m odd days. Thus 8 odd days = 1 odd day.**

**Lets understand the concept of finding odd days and the day on a particular week by some examples.**

**Example :- What was the day on 15 August 1947 ?**

**Solution :-** 15 august 1947 = (1946 years + Period from 1 january 1947 to 15 august 1947)

1600 years + 300 years + 47 years

1600 years = 0 odd days.

300 years = 1 odd day.

47 years = 11 leap year + 36 ordinary year

= 11 x 2 + 36 = 58 = 58/7 = 2 odd days.

Jan. Feb. March April May June July Aug.

31 + 28 + 31 + 30 + 31 + 30 + 31 + 15

= 227 days = 32 weeks + 3 days = 3 odd days

Total No. of odd days = 0+1+2+3 = 6 odd days

Hence , the required day was **Saturday.**

**Example :- What was the day on 16 April 2000 ?**

**Solution =** 16 April 2000 = (1999 year + Period from 1 january 2000 to 16 april 2000)

*Counting of odd days: *

1600 years + 300 years + 99 years

1600 years = 0 odd day.

300 year = 1 odd day

99 year = 24 leap year + 75 ordinary year

=(24 x 1 + 75 x 2) = 123 odd days = 123/7 = 4 odd days

Jan. Feb. March April

31 + 29 + 31 + 16

= 107 days = 15 weeks + 2 days = 2 odd days.

Total no. of odd days = 0+1+4+2 = 7 odd days = 0 odd days.

Hence, the required day was **Monday.**

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