Calendars: Solved Examples- 1

Question 1: If 15 March 1816 was Friday, what day of the week would 15th April 1916 be?
A. Monday
B. Wednesday
C. Thursday
D. Saturday

Ans: (d)

Sol: 15 March 1816 → 15 March 1916 is a span of 100 years. A period of 100 years has 5 odd days.
From 15 March 1916 to 15 April 1916: remaining days in March = 31 - 15 = 16 days; plus 15 days of April = 31 days → 31 days ≡ 3 odd days.
Total odd days = 5 + 3 = 8 ≡ 1 odd day.
Starting day Friday + 1 odd day = Saturday.

Question 2: Today is Monday. After 61 days, it will be:
A. Wednesday
B. Saturday
C. Tuesday
D. Thursday

Ans: (b)

Sol: The week repeats every 7 days.
61 days = 7 × 8 + 5 = 56 + 5, or observe 63 is multiple of 7 and 61 is 2 less than 63. Either way, 61 ≡ 5 (mod 7) or equivalently -2 (mod 7).
From Monday, moving forward 61 days is the same as moving forward 5 days: Monday → Tuesday(1) → Wednesday(2) → Thursday(3) → Friday(4) → Saturday(5).
Therefore, after 61 days it will be Saturday.

Question 3: If 28th August 1946 was a Wednesday, what day of the week was 31 August 1961?
A. Tuesday
B. Thursday
C. Monday
D. Wednesday

Ans: (b)

Sol: From 28 Aug 1946 to 28 Aug 1961 is 15 years. Count leap years in the period 1947-1960: 1948, 1952, 1956, 1960 → 4 leap years and 11 normal years.
Odd days = 11×1 + 4×2 = 11 + 8 = 19 ≡ 19 - 14 = 5 odd days.
From 28 Aug 1961 to 31 Aug 1961 = 3 days ≡ 3 odd days.
Total odd days = 5 + 3 = 8 ≡ 1 odd day.
Starting day Wednesday + 1 odd day = Thursday.

Question 4: If 09/12/2001(DD/MM/YYYY) happens to be Sunday, then 09/12/1971 would have been 
A. Wednesday
B. Tuesday
C. Saturday
D. Thursday

Ans: (d)

Sol: The span from 09/12/1971 to 09/12/2001 is 30 years. Count leap years between 1972 and 2000 (inclusive): 1972, 1976, 1980, 1984, 1988, 1992, 1996, 2000 → 8 leap years and 22 normal years.
Odd days = 22×1 + 8×2 = 22 + 16 = 38 ≡ 38 - 35 = 3 odd days.
So the weekday shifts forward by 3 days from 1971 to 2001. Given 09/12/2001 is Sunday, go back 3 odd days: Sunday - 3 = Thursday. Hence 09/12/1971 was Thursday.

Question 5: In 2016, Mohan celebrated his birthday on Friday. Which will be the first year after 2016 when Mohan will celebrate his birthday on a Wednesday? (He was not born in January or February)
A. 2021
B. 2023
C. 2020
D. 2025

Ans: (c)

Sol: For a birthday after February, a leap year contributes 2 odd days only if the leap year is the later year in the one-year interval (i.e., when the target year is a leap year). Count cumulative odd days from the birthday in 2016 onward until the weekday shifts from Friday to Wednesday. A shift from Friday to Wednesday is a net change of -2 days, which is equivalent to +5 odd days forward (because -2 ≡ +5 mod 7).
2017: +1 odd day → cumulative 1
2018: +1 → cumulative 2
2019: +1 → cumulative 3
2020: as 2020 is a leap year and is the later year in the interval 2019→2020, it contributes +2 → cumulative 5
Total 5 odd days achieved by 2020. Friday +5 odd days = Wednesday.
Therefore, the first year after 2016 when the birthday falls on Wednesday is 2020.

Question 6: What day of the week does May 28, 2006, fall on
A. Saturday 
B. Monday 
C. Sunday 
D. Thursday

Ans: (c)

Sol: Consider years completed before 2006 and days of 2006 up to 28 May.
Number of odd days in 2000 years = 0 (400-year cycle).
Odd days for 2001-2005: among these, 2004 is leap → 4 normal years + 1 leap = 4×1 + 1×2 = 6 odd days.
Days in 2006 from 1 Jan to 28 May = 31 (Jan) + 28 (Feb) + 31 (Mar) + 30 (Apr) + 28 (May) = 148 days = 21 weeks + 1 day → 1 odd day.
Total odd days = 0 + 6 + 1 = 7 ≡ 0 odd days.
0 odd days means the weekday is the same as 1 Jan of the base reference which here corresponds to Sunday. Hence 28 May 2006 was Sunday.

Question 7: What will be the day of the week on 15th August, 2010?
A. Thursday 
B. Sunday 
C. Monday 
D. Saturday

 Ans:  (b)

Sol: Count odd days from years and days in 2010 up to 15 Aug.
Odd days in 2001-2009: leap years are 2004 and 2008 → 7 normal + 2 leap = 7×1 + 2×2 = 11 ≡ 11 - 7 = 4 odd days.
Days from 1 Jan 2010 to 15 Aug 2010 = 31 + 28 + 31 + 30 + 31 + 30 + 31 + 15 = 227 days = 32 weeks + 3 days → 3 odd days.
Total odd days = 0 (2000-cycle) + 4 + 3 = 7 ≡ 0 odd days.
0 odd days corresponds to Sunday. Hence 15 August 2010 was Sunday.

Question 8: Today is Tuesday. After 52 days, it will be
A. Thursday 
B. Friday
C. Monday 
D. Saturday

Ans: (b)

Sol: The week repeats every 7 days. 52 = 7 × 7 + 3, so 52 ≡ 3 (mod 7).
From Tuesday, moving forward 3 days → Wednesday(1), Thursday(2), Friday(3).
Therefore, after 52 days it will be Friday.

Question 9: On what dates of April, 2001 did Wednesday fall?
A. 2^nd, 9^th, 16^th, 23^rd

B. 4^th, 11^th, 18^th, 25^th

C. 3^rd, 10^th, 17^th, 24^th

D. 1^st, 8^th, 15^th, 22^nd, 29^th

Ans:  (b)

Sol: Find weekday of 01-Apr-2001.
Odd days in 2000 years = 0.
Days from 1 Jan 2001 to 1 Apr 2001 = 31 (Jan) + 28 (Feb) + 31 (Mar) + 1 (Apr) = 91 days = 13 weeks → 0 odd days.
Total odd days = 0 + 0 = 0 → 01-Apr-2001 was Sunday.
Therefore Wednesdays in April 2001 fall on 4th, 11th, 18th and 25th. (Option B)

Question 10: How many days are there in x weeks x days
A. 14x
B. 8x

C. 7x²
D. 7

Ans:  (b)

Sol: x weeks x days = (7 × x) + x = 7x + x = 8x.

Question 11: The calendar for the year 2007 will be the same for the year
A. 2017
B. 2018
C. 2014
D. 2016

Ans: (b)

Sol: For a future year to have the same calendar as 2007, the total odd days accumulated from 2007 to that year must be ≡ 0 (mod 7). Count odd days year by year (2008 onwards) until the sum ≡ 0.
Odd days by year (2007→): 2008:+2, 2009:+1, 2010:+1, 2011:+1, 2012:+2, 2013:+1, 2014:+1, 2015:+1, 2016:+2, 2017:+1. Summing these from 2008 through 2017 gives 14 odd days ≡ 0.
Thus the calendar of 2018 will match that of 2007.

Question 12: Which of the following is not a leap year?
A. 1200
B. 800
C. 700
D. 2000

Ans:  (c)

Sol: Leap year rules:
1. A non-century year divisible by 4 is a leap year.
2. A century year (ending with 00) is a leap year only if it is divisible by 400.
Years 800, 1200 and 2000 are divisible by 400, so they are leap years.
Year 700 is a century year but not divisible by 400 → it is not a leap year.

Question 13: 01-Jan-2007 was Monday. What day of the week lies on 01-Jan-2008?
A. Wednesday
B. Sunday
C. Friday
D. Tuesday

Ans:  (d)

Sol: The whole year 2007 is a non-leap year → 1 odd day for the year.
01-Jan-2008 = Monday + 1 odd day = Tuesday.

Question 14: 8th Dec 2007 was Saturday, what day of the week was it on 8th Dec, 2006?
A. Sunday
B. Tuesday
C. Friday
D. Tuesday

Ans:  (c)

Sol: From 8 Dec 2006 to 7 Dec 2007 is 365 days = 1 odd day (2007 is not between leap Feb here).
Therefore 8 Dec 2006 = Saturday - 1 odd day = Friday.

Question 15: On 8th Feb, 2005 it was Tuesday. What was the day of the week on 8th Feb, 2004?
A. Sunday
B. Friday
C. Saturday
D. Monday

Ans:  (a)

Sol: Interval from 8 Feb 2004 to 7 Feb 2005 includes 29 Feb 2004 because 2004 is a leap year; hence that interval has 366 days = 2 odd days.
So 8 Feb 2004 = 8 Feb 2005 - 2 odd days = Tuesday - 2 = Sunday.