Calendars questions with answers and concepts

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112 practice questionsPage updated 11 Nov 2026Review metadata pendingPractice questions

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Calendar problems involve determining the day of the week for a given date, calculating odd days, and understanding leap year rules. These problems test your ability to work with the Gregorian calendar system and apply modular arithmetic concepts. Mastering calendar calculations is essential for solving date-based logical reasoning questions efficiently.

Key formulas and rules

7 rules

Odd Days formula: $Odd Days = Total Days (mod 7)$

Normal Year: $365 days = 52 weeks + 1 day ⟹ 1 odd day$

Leap Year: $366 days = 52 weeks + 2 days ⟹ 2 odd days$ (divisible by $4$ , century years must divide by $400$ )

Century Odd Days: $100 yrs = 5 odd days$ , $200 yrs = 3$ , $300 yrs = 1$ , $400 yrs = 0$

Day Codes: $Sun = 0$ , $Mon = 1$ , $Tue = 2$ , $Wed = 3$ , $Thu = 4$ , $Fri = 5$ , $Sat = 6$

Calendar Cycle: The Gregorian calendar repeats exactly every $400 years$

Zeller's Congruence: $h = (q + ⌊ \frac{13 ( m + 1 )}{5} ⌋ + K + ⌊ \frac{K}{4} ⌋ + ⌊ \frac{J}{4} ⌋ - 2 J) (mod 7)$ (where $q = day$ , $m = month with March = 3 \dots Feb = 14$ , $K = year mod 100$ , $J = ⌊ year /100 ⌋$ ; in Zeller's convention $h = 0 ⟹ Saturday, 1 ⟹ Sunday, \dots, 6 ⟹ Friday$ )

Key concepts

Odd Days Concept

Odd days are the extra days left after complete weeks are removed from any given period. A normal year has 365 days = 52 weeks + 1 day, giving 1 odd day. A leap year has 366 days = 52 weeks + 2 days, giving 2 odd days. To find the day of week, calculate total odd days and apply modulo 7. For example, if today is Monday and we add 3 odd days, the result day is Thursday.

Leap Year Rules

A year is a leap year if: (1) divisible by 4 but not by 100, OR (2) divisible by 400. Years like 1900, 2100 are NOT leap years despite being divisible by 4. Years like 2000, 2400 ARE leap years. February has 29 days in leap years and 28 in normal years. This distinction is crucial for calculating odd days across February boundaries.

Century Calculations

Every 100 years, the calendar shifts by 5 odd days (since 76 normal years x 1 + 24 leap years x $2 = 124$ odd days = 5 mod 7). For multiple centuries: 100 years = 5 odd days, 200 years = 3 odd days ( $5 + 5 = 10$ , 10 mod $7 = 3$ ), 300 years = 1 odd day, 400 years = 0 odd days (accounting for the leap century). This pattern repeats every 400 years.

Month Code System

To quickly calculate day shifts in standard month-code arithmetic: January=0 (or 6 in a leap year), February=3 (or 2 in a leap year), March=3, April=6, $M a y = 1$ , June=4, July=6, August=2, September=5, October=0, November=3, December=5. These codes represent the cumulative odd-day offset from the start of the year. Add the date number and apply modulo 7 to determine the weekday.

Same Day Date Patterns and Repetition

Calendar date alignments (when the 1st of the month falls on the same weekday):
- In an ordinary year ( $F e b = 28$ days): Jan & Oct; Feb, Mar & Nov; Apr & Jul; Sep & Dec.
- In a leap year ( $F e b = 29$ days): Jan, Apr & Jul; Feb & Aug; Mar & Nov; Sep & Dec.

Calendar repetition rules:
(1) A leap year calendar repeats every 28 years (within a single century).
(2) An ordinary year immediately following a leap year repeats after 6 years (e.g. 2005 to 2011) because the 6-year span contains exactly 1 leap year ( $5 \times 1 + 1 \times 2 = 7 \equiv 0 (mod 7)$ ).
(3) Other ordinary years repeat after 11 years (e.g. 2006 to 2017) because a 6-year span contains 2 leap years ( $4 \times 1 + 2 \times 2 = 8 \neq \equiv 0$ ).

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Worked examples

Example 1

What day of the week was 15th August 1947?

Solution

Step 1: Decompose completed years (1946 completed years).
- $1600 years = 0 odd days$ (multiple of 400)
- $300 years = 1 odd day$
- $46 years$ : Number of leap years = $⌊ 46/4 ⌋ = 11$ ; ordinary years = $46 - 11 = 35$

Odd days in 46 years = (11 \times 2) + (35 \times 1) = 22 + 35 = 57 \equiv 1 (mod 7)

Step 2: Compute odd days in 1947 up to August 15.
- Jan (31): 3, Feb (28): 0, Mar (31): 3, Apr (30): 2, May (31): 3, Jun (30): 2, Jul (31): 3, Aug (15): $15 \equiv 1$

Sum = 3 + 0 + 3 + 2 + 3 + 2 + 3 + 1 = 17 \equiv 3 (mod 7)

Step 3: Total odd days.

Total = 0 + 1 + 1 + 3 = 5 (mod 7)

Odd day code 5 corresponds to Friday.

Answer: Friday.

Example 2

If 1st January 2006 was a Sunday, what day of the week was 1st January 2010?

Solution

Step 1: Count years elapsed from 2006 to 2010.
- 4 years: 2006, 2007, 2008, 2009
- 2008 is a leap year (2 odd days); 2006, 2007, and 2009 are ordinary (1 odd day each)

Step 2: Sum odd days.

Total Odd Days = 1 + 1 + 2 + 1 = 5 days

Step 3: Advance 5 days from Sunday.

Sunday + 5 = Friday

Answer: Friday.

Example 3

Which upcoming calendar year will have the exact same calendar as the year 2024?

Solution

Step 1: Check leap year cycle.
2024 is a leap year. A leap year calendar repeats only on another leap year.

Step 2: Apply the 28-year repetition rule for leap years.

2024 + 28 = 2052

Answer: 2052.

Example 4

What is the maximum number of times the 13th of a month can fall on a Friday in any single calendar year?

Solution

Step 1: A Friday the 13th occurs whenever the 1st of that month is a Sunday.

Step 2: Analyze the distribution of first-of-month days.
In a standard or leap year, at most three months can start on the same weekday (e.g., in a non-leap year starting on Thursday: February, March, and November).

Answer: 3 times.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

How many odd days are there in 100 years?

Answer: B. 5

Explanation

Step 1: In 100 years, leap years = 24 and ordinary years = 76.
Step 2: Odd days = 76x1 + 24x2 = 76 + 48 = 124.
Step 3: 124 mod $7 = 5$ .
Answer: 5

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

How many odd days are there in 200 years?

Answer: A. 3

Explanation

Step 1: 200 years = two blocks of 100 years.
Step 2: Each 100 years contributes 5 odd days.
Step 3: Total = 10 odd days.
Step 4: 10 mod $7 = 3$ .
Answer: 3

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

How many odd days are there in 300 years?

Answer: B. 1

Explanation

Step 1: 300 years = three blocks of 100 years.
Step 2: Each block has 5 odd days.
Step 3: Total = 15 odd days.
Step 4: 15 mod $7 = 1$ .
Answer: 1

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Always verify leap years carefully: century years (1900, 2100) are NOT leap years unless divisible by 400 (2000, 2400 are leap years).
Use the century odd days pattern: 100=5, 200=3, 300=1, 400=0. This repeats every 400 years, making calendar calculations cyclical.
For quick month calculations, memorize that Jan+Feb combined have 59 days (60 in leap), which is 3 or 4 odd days respectively.
When finding a past date's day, subtract odd days instead of adding, or add (7 - odd days) to move backward equivalently.
Same day of week falls on: 1 Jan (non-leap) = 1 Oct; 1 Jan (leap) = 1 Jul; 29 Feb only exists in leap years.
Use reference dates you know: Independence Day 1947 = Friday, Republic Day 1950 = Thursday, or any known date as anchor point.

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Key concepts

Odd Days Concept

Leap Year Rules

Century Calculations

Month Code System

Same Day Date Patterns and Repetition

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 1

What day of the week was 15th August 1947?

Solution

Odd days in 46 years = (11 \times 2) + (35 \times 1) = 22 + 35 = 57 \equiv 1 (mod 7)

Step 2: Compute odd days in 1947 up to August 15.
- Jan (31): 3, Feb (28): 0, Mar (31): 3, Apr (30): 2, May (31): 3, Jun (30): 2, Jul (31): 3, Aug (15): $15 \equiv 1$

Sum = 3 + 0 + 3 + 2 + 3 + 2 + 3 + 1 = 17 \equiv 3 (mod 7)

Step 3: Total odd days.

Total = 0 + 1 + 1 + 3 = 5 (mod 7)

Odd day code 5 corresponds to Friday.

Answer: Friday.

Example 2

If 1st January 2006 was a Sunday, what day of the week was 1st January 2010?

Solution

Step 1: Count years elapsed from 2006 to 2010.
- 4 years: 2006, 2007, 2008, 2009
- 2008 is a leap year (2 odd days); 2006, 2007, and 2009 are ordinary (1 odd day each)

Step 2: Sum odd days.

Total Odd Days = 1 + 1 + 2 + 1 = 5 days

Step 3: Advance 5 days from Sunday.

Sunday + 5 = Friday

Answer: Friday.

Example 3

Which upcoming calendar year will have the exact same calendar as the year 2024?

Solution

Step 1: Check leap year cycle.
2024 is a leap year. A leap year calendar repeats only on another leap year.

Step 2: Apply the 28-year repetition rule for leap years.

2024 + 28 = 2052

Answer: 2052.

Example 4

What is the maximum number of times the 13th of a month can fall on a Friday in any single calendar year?

Solution

Step 1: A Friday the 13th occurs whenever the 1st of that month is a Sunday.

Answer: 3 times.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

How many odd days are there in 100 years?

Answer: B. 5

Explanation

Step 1: In 100 years, leap years = 24 and ordinary years = 76.
Step 2: Odd days = 76x1 + 24x2 = 76 + 48 = 124.
Step 3: 124 mod $7 = 5$ .
Answer: 5

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

How many odd days are there in 200 years?

Answer: A. 3

Explanation

Step 1: 200 years = two blocks of 100 years.
Step 2: Each 100 years contributes 5 odd days.
Step 3: Total = 10 odd days.
Step 4: 10 mod $7 = 3$ .
Answer: 3

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

How many odd days are there in 300 years?

Answer: B. 1

Explanation

Step 1: 300 years = three blocks of 100 years.
Step 2: Each block has 5 odd days.
Step 3: Total = 15 odd days.
Step 4: 15 mod $7 = 1$ .
Answer: 1

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Always verify leap years carefully: century years (1900, 2100) are NOT leap years unless divisible by 400 (2000, 2400 are leap years).
Use the century odd days pattern: 100=5, 200=3, 300=1, 400=0. This repeats every 400 years, making calendar calculations cyclical.
For quick month calculations, memorize that Jan+Feb combined have 59 days (60 in leap), which is 3 or 4 odd days respectively.
When finding a past date's day, subtract odd days instead of adding, or add (7 - odd days) to move backward equivalently.
Same day of week falls on: 1 Jan (non-leap) = 1 Oct; 1 Jan (leap) = 1 Jul; 29 Feb only exists in leap years.
Use reference dates you know: Independence Day 1947 = Friday, Republic Day 1950 = Thursday, or any known date as anchor point.

Key formulas and rules

Key concepts

Odd Days Concept

Leap Year Rules

Century Calculations

Month Code System

Same Day Date Patterns and Repetition

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.

Key formulas and rules

Key concepts

Odd Days Concept

Leap Year Rules

Century Calculations

Month Code System

Same Day Date Patterns and Repetition

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.