Calendar
Leap Year :(1)= A particular year divisible by 4 is called leap year, also it should not be century year
Eg. 1976, 1840 are leap year as they are divisible by 4
(2)= Each fourth century is a leap year
Eg. 400, 800, 1200, 1600, 2000, 2400 are leap years, but 700, 1300, 1900 are not leap years
Odd Days : For a given number of days, number of days more than complete week are called odd days.
Eg. in 10 days, there is one week and 3 odd days.
Counting of weeks and days in year:
(a) 1 ordinary year has 365 days = 52 weeks + 1 odd day
(b) 1 leap year has 366 days = 52 weeks + 2 odd days
(c) 100 years = 76 ordinary years + 24 leap years = 76*1 + 24*2 = 124 odd days
= 17 weeks + 5 odd days
(d) Since in 100 years, number of odd days are 5
(e) In 200 years, number of odd days are 5*2= 10 = 1 week + 3 odd days, 3 odd days
(f) Similarly in 300 years 5*3 = 15 = 2 weeks + 1 odd day
(g) Similarly for 400 years number of odd days 5*4 + 1 = 0 odd days, from this we conclude that for 800, 1200, 1600 , 2000, 2400 years odd days will be zero
| Day | Sun | Mon | Tues | Wed | Thurs | Fri | Sat |
| Odd day no. | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
Ex. What was the day of the week on 20 may, 1985 ?
Solution : Here Number of odd days in 1600 years = 0
Number of odd days in 300 years from 1600 to 1900 = 5*3 = 2 week + 1 odd day= 1 odd day
Number of odd days in 84 years= 21 leap year + 63 days = 21*2 + 63*1 = 105 days = 0 odd days
Number of odd days in 20 may = 31 days of Jan. + 28 days of Feb + 31 days of mar. + 30 days in April + 20 days in may = 140 days = 0 odd day
So total number of odd days = 0+1+0+0=1 = Monday
Ex. Prove that calendar for the year 2009 will serve for the year 2015.
Solution : for this sum of odd days from 2009 to 2014 should be zero.
| Year | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 |
| Odd day | 1 | 1 | 1 | 2 | 1 | 1 |
Sum of odd days = 1+1+1+2+1+1=7= 1 week + 0 odd days
So, both dates 1.1.2009 and 1.1.2015 will be on same day , so calendar for the year 2009 will serve for the year 2015
Ex. On what date of Feb. 2007 did Saturday fall ?
Solution : For this find the day of 1.2.2007
1600+400 years has 0 odd days
From 2001 to 2006 there are 1 leap years + 5 ordinary years
So number of odd days = 1*2 + 5*1 = 2 + 5 = 7 = 1 week = 0 odd day
Now from 1.1.2007 to 1.2.2007 number of days = 32 = 4 weeks + 4 odd days = 4 odd days
So, total number of odd days = 4, so 1.2.2007 will be Thursday
Now Saturday will be on 3.2.2007
Ex. Today is Tuesday. After 72 days, it will be ?
Solution : Tuesday will be repeated after each 7 days so, after at 70 day it will also be Tuesday, so at 72 th day it will be Friday
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