Calendar / Date & Day **** Calendar of the year "xxxx" will repeat in which year ?

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Rule to Find Leap Year

A year is a leap year if it satisfies either of these conditions: i. Divisible by 4 but not by 100,  ii. Divisible by 400

1. Check if the year is divisible by 4 : i. If no, then not a leap year,  ii. If yes, check divisible by 100 or not

2. Check if the year is divisible by 100 : i. If no, it is a leap year,  If yes, check if divisible by 400

i. If yes, it is a leap year,  ii. If no, not a leap year

Normal Year

• A normal year has 365 days.


• 365 ÷ 7 = 52 weeks + 1 day leftover


• That leftover 1 day shifts the weekday forward by 1 next year.

Ex: If 01.01. 2020 is Wednesday, 1 Jan 2021 → Thursday (+1 day)

Leap Year

• A leap year has 366 days.


• 366 ÷ 7 = 52 weeks + 2 days leftover


• That leftover 2 days shifts the weekday forward by 2 next year.

Ex: If 01.01. 2016 is Friday, 01.01. 2017 → Sunday (+2 days)

Quick Rule

• Normal year → +1 day


• Leap year → +2 days

This is why we add 1 or 2 days for each year when calculating the day of the week.

Examples :

Q. If 31 August 2000 was Friday, then which day was 31 August 2009 ?

Options: (a) Monday (b) Tuesday (c) Wednesday (d) Friday

Solⁿ:

1. Count the leap years between 2000 and 2009 → 2004, 2008 → each gives +2 days


2. Count the normal years → 2001, 2002, 2003, 2005, 2006, 2007, 2009 → each gives +1 day

    Total days added: 2 + 2 + 7 = 11 → 11 ÷ 7 = 1 remainder 4

     Move 4 days forward from Friday: Friday → Saturday → Sunday → Monday → Tuesday

Ans: Tuesday

Q. If 1 August 2009 was Saturday, then which day was 1 August 2013 ?

Options: (a) Saturday (b) Thursday (c) Friday (d) Sunday

Solⁿ:  

1. Count years from 2009 → 2013:

    2010, 2011, 2013 → normal years → +1 day each

    2012 → leap year → +2 days

Total shift: 1+1+2+1 = 5 days

1. Move 5 days forward from Saturday:


2. Saturday → Sunday → Monday → Tuesday → Wednesday → Thursday

Ans: Thursday

Q. If 28 February 2017 was Tuesday, then which day was 28 February 2019 ?

Options: (a) Thursday (b) Saturday (c) Friday (d) Monday

Solⁿ:  

Count years from 2017 → 2019: 2018 :  normal → +1 day, 2019 : normal → +1 day

Total shift: 1 + 1 = 2 days

Move 2 days forward from Tuesday: Tuesday → Wednesday → Thursday

Ans: Thursday

Q. If 15 March 1996 was Friday, then which day was 15 March 2023 ?

Options: (a) Wednesday (b) Thursday (c) Friday (d) Saturday

Solⁿ:

1. Count leap years 1996 → 2023 → 1996, 2000, 2004, 2008, 2012, 2016, 2020 → 7 leap years (+2 each)


2. Normal years → 28 – 7 = 21 (+1 each)

Total shift: 7×2 + 21×1 = 14 + 21 = 35

Divide by 7 → remainder 0 → no change in day

Ans: Friday

2. Odd Days

Odd days = Extra days left after making complete weeks

Odd Days Summary

i. 1 Ordinary Year → 1 odd day

ii. 1 Leap Year → 2 odd days

iii. 100 Years → 5 odd days

iv. 200 Years → 3 odd days

v. 300 Years → 1 odd day

vi. 400 Years → 0 odd days (calendar repeats)

After every 400 years, the calendar repeats exactly.

3. Month-wise Odd Days

• January → 3 odd days (same in leap year)


• February → 0 odd days (→ 1 in leap year)


• March → 3 odd days


• April → 2 odd days


• May → 3 odd days


• June → 2 odd days


• July → 3 odd days


• August → 3 odd days


• September → 2 odd days


• October → 3 odd days


• November → 2 odd days


• December → 3 odd days

Only February changes in a leap year (0 → 1).

4. Day Codes (0–6 Method)

• Sunday → 0


• Monday → 1


• Tuesday → 2


• Wednesday → 3


• Thursday → 4


• Friday → 5


• Saturday → 6

These codes are used to calculate the exact day of any date.

5. Formula to Find Day of Any Date

Day = (Year Code + Month Code + Date + Century Code) mod 7

Meaning (by point)

• Year Code → calculated from last two digits of the year


• Month Code → fixed code for each month


• Date → given date (1 to 31)


• Century Code → depends on century (1600, 1700, etc.)


• mod 7 → divide total by 7 and take remainder

The remainder gives the day code (0–6)

Then match with day codes: 0 = Sunday, 1 = Monday … 6 = Saturday

Century Codes: 1600 → 6, 1700 → 4, 1800 → 2, 1900 → 0, 2000 → 6

These codes help calculate the day for any date in that century.

Year Code (How to Calculate)

1. Take last two digits of the year = N


2. Year Code = N + ⌊N/4⌋

Ex: Year = 1987

• Last two digits = 87


• Year Code = 87 + ⌊87/4⌋


• = 87 + 21


• = 108

6. Key Concepts

A. Same Calendar Condition

Two years will have the same calendar when:

• Both are leap years or both are non - leap years


• Odd day difference = 0

Ex: → 2017 and 2023 have the same calendar

B. Day Forward / Backward Rule

• After N days: (X+N) mod 7  


• Before N days: (X−N) mod 7

C. Calendar Repeat

• Normal year → repeats after 6 or 11 years


• Leap year → repeats after 28 years

Means the exact same dates + same days will match again after this period.

Exam Tricks

1. After 100 days

100 mod 7 = 2 

Day will be 2 days ahead

2. Same calendar as 2025

i. 2025 is normal year

ii. Odd day = 1

iii. Next matching calendar = 2031

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3. Find the Day: 15 August 1947

i. Century code → 0

ii. Last two digits → 47

iii. Year code → 47 + 11 = 58

iv. Month code (August) → 3

v. Date → 15

Total Calculation : 0+58+3+15=76

         76 mod 7 = 6

Day Code Result : Day code 6 = Friday

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Q. If 1 January is Friday, then 1 February will be which day ? প্ৰশ্ন: যদি ১ জানুৱাৰী শুক্ৰবাৰ হয়, তেন্তে ১ ফেব্ৰুৱাৰী কি বাৰ হ’ব ?

Solution / সমাধান:

January has 31 days. (জানুৱাৰী মাহত ৩১ দিন থাকে।)

$$31÷7=4\text{weeks }+3\text{ days}$$

So, there are 3 extra days. অৰ্থাৎ ৩ দিন অতিৰিক্ত থাকে।

Starting from Friday / শুক্ৰবাৰৰ পৰা গণনা:

Friday → Saturday → Sunday → Monday (শুক্ৰবাৰ → শনিবাৰ → দেওবাৰ → সোমবাৰ)

Therefore, 1 February = Monday

Ans / উত্তৰ: Monday / সোমবাৰ

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Q. 58 days ago was Tuesday. What day is today ?

Trick:

58 ÷ 7 = 8 weeks remainder 2

So, 58 days ago = 2 days before today.

• 2 days ago → Tuesday, 1 day ago → Wednesday, Today → Thursday

Ans: Thursday

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Q. 14 July = Tuesday. What day will be 24 October ?

Solution:

Days from 14 July to 24 October:

• July: 31 − 14 = 17 days


• August = 31 days


• September = 30 days


• October = 24 days

Total = 17 + 31 + 30 + 24 = 102 days

Now divide by 7: 102 ÷ 7 = 14 weeks + 4 days

So, move 4 days forward from Tuesday:

Tuesday → Wednesday → Thursday → Friday → Saturday

Ans: Saturday

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Q. 45 days from now will be Saturday. What day is today ?

Solution:

$$45÷7=6\text{ weeks }+3\text{ days}$$

So, 45 days later = 3 days ahead.

If 45 days later is Saturday, go 3 days backward:

Saturday → Friday → Thursday → Wednesday

Ans: Wednesday (B)

Trick: Divide by 7 → Take remainder → Count backward from the given day.

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Q. 50 days ago was Sunday. What day is today ?

Solution:

$$50÷7=7\text{weeks }+1\text{ day}$$

So, today is 1 day after Sunday:

Sunday → Monday

Ans: Monday

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