Calender

1. 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. 2. Leap Year: (i). Every year divisible by 4 is a leap year, if it is not a century.   (ii). Every 4^{th} century is a leap year and no other century is a leap year. Note: A leap year has 366 days. Examples: i. Each of the years 1948, 2004, 1676 etc. is a leap year. ii. Each of the years 400, 800, 1200, 1600, 2000 etc. is a leap year. iii. None of the years 2001, 2002, 2003, 2005, 1800, 2100 is a leap year.   3. Ordinary Year:’   The year which is not a leap year is called an ordinary years. An ordinary year has 365 days. 4. Counting of Odd Days: i. 1 ordinary year = 365 days = (52 weeks + 1 day.) ∴   1 ordinary year has 1 odd day. ii. 1 leap year = 366 days = (52 weeks + 2 days) ∴    1 leap year has 2 odd days. iii. 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. ∴  Number of odd days in 100 years = 5. Number of odd days in 200 years = (5 x 2) 3 odd days. Number of odd days in 300 years = (5 x 3) 1 odd day. 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. 5. Day of the Week Related to Odd Days:
No. of days:0123456
Day:Sun.Mon.Tues.Wed.Thurs.Fri.Sat.
1. It was Sunday on Jan 1, 2006. What was the day of the week Jan 1, 2010?        A. Sunday        B. Saturday        C. Friday        D. Wednesday       Answer: Option C       Explanation: On 31st December, 2005 it was Saturday. Number of odd days from the year 2006 to the year 2009 = (1 + 1 + 2 + 1) = 5 days. ∴ On 31st December 2009, it was Thursday. Thus, on 1st Jan, 2010 it is Friday. 2.    What was the day of the week on 28th May, 2006? A. Thursday B. Friday  C. Saturday D. Sunday Answer: Option D Explanation: 28 May, 2006 = (2005 years + Period from 1.1.2006 to 28.5.2006) Odd days in 1600 years = 0 Odd days in 400 years = 0 5 years = (4 ordinary years + 1 leap year) = (4 x 1 + 1 x 2) 6 odd days Jan. Feb. March April May (31 + 28 + 31 + 30 + 28 ) = 148 days 148 days = (21 weeks + 1 day) 1 odd day. Total number of odd days = (0 + 0 + 6 + 1) = 7 0 odd day. Given day is Sunday. 3. What was the day of the week on 17th June, 1998? A. Monday B. Tuesday  C. Wednesday D. Thursday Answer: Option C Explanation: 17^{th} June, 1998 = (1997 years + Period from 1.1.1998 to 17.6.1998) Odd days in 1600 years = 0 Odd days in 300 years = (5 x 3) 1 97 years has 24 leap years + 73 ordinary years. Number of odd days in 97 years ( 24 x 2 + 73) = 121 = 2 odd days. Jan. Feb. March April May June (31 + 28 + 31 + 30 + 31 + 17) = 168 days ∴ 168 days = 24 weeks = 0 odd day. Total number of odd days = (0 + 1 + 2 + 0) = 3. Given day is Wednesday. 4. What will be the day of the week 15^{th}August, 2010? A. Sunday   B. Monday C. Tuesday  D. Friday Answer: Option A Explanation: 15^{th} August, 2010 = (2009 years + Period 1.1.2010 to 15.8.2010) Odd days in 1600 years = 0 Odd days in 400 years = 0 9 years = (2 leap years + 7 ordinary years) = (2 x 2 + 7 x 1) = 11 odd days 4 odd days. Jan. Feb. March April May June July Aug. (31 + 28 + 31 + 30 + 31 + 30 + 31 + 15) = 227 days ∴ 227 days = (32 weeks + 3 days) 3 odd days. Total number of odd days = (0 + 0 + 4 + 3) = 7 0 odd days. Given day is Sunday. 5. Today is Monday. After 61 days, it will be:  A. Wednesday  B. Saturday  C. Tuesday  D. Thursday Answer: Option B Explanation: Each day of the week is repeated after 7 days. So, after 63 days, it will be Monday. ∴ After 61 days, it will be Saturday. 6. If 6th March, 2005 is Monday, what was the day of the week on 6th March, 2004? A. Sunday B. Saturday C. Tuesday D. Wednesday Answer: Option A Explanation: The year 2004 is a leap year. So, it has 2 odd days. But, Feb 2004 not included because we are calculating from March 2004 to March 2005. So it has 1 odd day only. ∴The day on 6^{th} March, 2005 will be 1 day beyond the day on 6th March, 2004. Given that, 6^{th} March, 2005 is Monday. ∴ 6^{th} March, 2004 is Sunday (1 day before to 6^{th} March, 2005). 7. On what dates of April, 2001 did Wednesday fall? A. 1^{st}, 8^{ht}, 15^{th}, 22^{nd}, 29^{th} B. 2^{nd}, 9^{th}, 16^{th}, 23^{rd}, 30^{th}  C. 3^{rd}, 10^{th}, 17^{th}, 24^{th}  D. 4^{th}, 11^{th}, 18^{th}, 25^{th} Answer: Option D Explanation: We shall find the day on 1^{st} April, 2001. 1^{st} April, 2001 = (2000 years + Period from 1.1.2001 to 1.4.2001) Odd days in 1600 years = 0 Odd days in 400 years = 0 Jan. Feb. March April (31 + 28 + 31 + 1) = 91 days 0 odd days. Total number of odd days = (0 + 0 + 0) = 0 On 1^{st} April, 2001 it was Sunday. In April, 2001 Wednesday falls on 8^{th},11^{th}, 18^{th} and 25^{th}. 8.   How many days are there in x weeks x days? A. 7x^{2} B. 8x C.14x  D. 7  Answer: Option B Explanation: x weeks x days = (7x + x) days = 8x days. 9.   The last day of a century cannot be A. Monday B. Wednesday   C. Tuesday   D. Friday Answer: Option C   Explanation: 100 years contain 5 odd days. ∴Last day of 1st century is Friday. 200 years contain (5 x 2) 3 odd days. ∴Last day of 2nd century is Wednesday. 300 years contain (5 x 3) = 15 1 odd day. ∴Last day of 3rd century is Monday. 400 years contain 0 odd day. ∴Last day of 4th century is Sunday. This cycle is repeated. ∴Last day of a century cannot be Tuesday or Thursday or Saturday.   10. On 8^{th} Feb, 2005 it was Tuesday. What was the day of the week on 8^{th} Feb, 2004?  A. Tuesday B. Monday C. Sunday D. Wednesday Answer: Option C Explanation: The year 2004 is a leap year. It has 2 odd days. ∴   The day on 8th Feb, 2004 is 2 days before the day on 8th Feb, 2005. Hence, this day is Sunday. 11. The calendar for the year 2007 will be the same for the year: A. 2014 B. 2016 C. 2017 D. 2018 Answer: Option D Explanation: Count the number of odd days from the year 2007 onwards to get the sum equal to 0 odd day.
Year :200720082009  201020112012 20132014201520162017
Odd day :12111211121
Sum = 14 odd days 0 odd days. ∴ Calendar for the year 2018 will be the same as for the year 2007. ∴ 12. Which of the following is not a leap year?  A. 700 B. 800 C. 1200 D. 2000 Answer: Option A    Explanation: The century divisible by 400 is a leap year. ∴The year 700 is not a leap year. 13. On 8^{th} Dec, 2007 Saturday falls. What day of the week was it on 8^{th}Dec, 2006?  A. Sunday B. Thursday C. Tuesday  D. Friday Answer: Option D  Explanation: The year 2006 is an ordinary year. So, it has 1 odd day. So, the day on 8^{th} Dec, 2007 will be 1 day beyond the day on 8^{th}Dec, 2006.But, 8^{th} Dec, 2007 is Saturday. ∴ 8^{th} Dec, 2006 is Friday. 14. January 1, 2008 is Tuesday. What day of the week lies on Jan 1, 2009? A. Monday B. Wednesday C. Thursday D. Sunday Answer: Option C Explanation: The year 2008 is a leap year. So, it has 2 odd days. 1^{st} day of the year 2008 is Tuesday (Given) So, 1^{st}day of the year 2009 is 2 days beyond Tuesday. Hence, it will be Thursday. 15. January 1, 2007 was Monday. What day of the week lies on Jan. 1, 2008? A. Monday B. Tuesday  C. Wednesday D. Sunday Answer: Option B Explanation: The year 2007 is an ordinary year. So, it has 1 odd day. 1^{st}day of the year 2007 was Monday. 1^{st} day of the year 2008 will be 1 day beyond Monday. Hence, it will be Tuesday.