If Republic Day was celebrated in 1996 on Friday, on which day was Independence Day celebrated in 2000?

Question

Prepp Team · Accepted Answer

Understanding the Calendar Calculation Problem

This question asks us to determine the day of the week for Independence Day (August 15th) in the year 2000, given that Republic Day (January 26th) in 1996 was on a Friday. To solve this, we need to calculate the total number of "odd days" between the two dates. Odd days are the extra days remaining after dividing the total number of days by 7 (since a week has 7 days).

Key Concepts: Leap Years and Odd Days

Leap Year: A year that has 366 days instead of the usual 365. An extra day is added in February (February 29th). A year is a leap year if it is divisible by 4, except for century years which must be divisible by 400. Both 1996 and 2000 are leap years.
Odd Days: The remainder when the total number of days is divided by 7. The day of the week repeats every 7 days. If we know the day of the week on a certain date, we can find the day of the week on another date by adding the number of odd days between them to the starting day.

Step-by-Step Calculation

We need to find the number of odd days from January 26, 1996, to August 15, 2000. We can break this down into two parts:

Calculate the number of odd days from January 26, 1996, to January 26, 2000.
Calculate the number of odd days from January 26, 2000, to August 15, 2000.

Part 1: Odd Days from Jan 26, 1996 to Jan 26, 2000

We will look at the period year by year starting from Jan 26th of each year.

From Jan 26, 1996 to Jan 26, 1997: This period includes February 1996 (a leap year). So, it is 366 days. Number of odd days = $\frac{366}{7} = 52 \text{ weeks} + 2 \text{ days}$. Odd days = 2.
From Jan 26, 1997 to Jan 26, 1998: This is a normal year (365 days). Number of odd days = $\frac{365}{7} = 52 \text{ weeks} + 1 \text{ day}$. Odd days = 1.
From Jan 26, 1998 to Jan 26, 1999: This is a normal year (365 days). Number of odd days = 1.
From Jan 26, 1999 to Jan 26, 2000: This is a normal year (365 days). Number of odd days = 1.

Total odd days from Jan 26, 1996 to Jan 26, 2000 = $2 + 1 + 1 + 1 = 5$ odd days.

This means Jan 26, 2000, was Friday + 5 days, which is Wednesday.

Part 2: Odd Days from Jan 26, 2000 to Aug 15, 2000

Now we calculate the number of days remaining in 2000 from Jan 26 up to Aug 15. Remember that 2000 is a leap year, so February has 29 days.

January: $31 - 26 = 5$ days
February: 29 days (2000 is a leap year)
March: 31 days
April: 30 days
May: 31 days
June: 30 days
July: 31 days
August: 15 days (up to Independence Day)

Total number of days from Jan 26, 2000 to Aug 15, 2000 = $5 + 29 + 31 + 30 + 31 + 30 + 31 + 15 = 202$ days.

Now, find the number of odd days in 202 days: $\frac{202}{7}$.

$202 = 7 \times 28 + 6$. The remainder is 6. So, there are 6 odd days.

Alternatively, we can calculate odd days for each month period:

Month (2000)	Number of Days	Odd Days ($\text{Days} \pmod{7}$)
Jan (remaining)	5	$5 \pmod{7} = 5$
Feb	29	$29 \pmod{7} = 1$
Mar	31	$31 \pmod{7} = 3$
Apr	30	$30 \pmod{7} = 2$
May	31	$31 \pmod{7} = 3$
Jun	30	$30 \pmod{7} = 2$
Jul	31	$31 \pmod{7} = 3$
Aug (up to 15)	15	$15 \pmod{7} = 1$

Total odd days from Jan 26, 2000 to Aug 15, 2000 = $5 + 1 + 3 + 2 + 3 + 2 + 3 + 1 = 20$ odd days.

Number of odd days in 20 = $20 \pmod{7} = 6$ odd days.

Both methods give 6 odd days for this period.

Part 3: Combining Odd Days and Finding the Final Day

Total odd days from Jan 26, 1996 to Aug 15, 2000 = (Odd days from Part 1) + (Odd days from Part 2)

Total odd days = $5 + 6 = 11$ odd days.

Number of effective odd days = $11 \pmod{7} = 4$ odd days.

The starting day was Friday. We need to add 4 odd days to Friday:

Friday + 1 day = Saturday
Saturday + 1 day = Sunday
Sunday + 1 day = Monday
Monday + 1 day = Tuesday

So, Independence Day in 2000 was on a Tuesday.

Summary of Calendar Day Calculation

By calculating the total number of odd days between Republic Day 1996 and Independence Day 2000, we found the shift in the day of the week. The total period resulted in 4 effective odd days. Adding these 4 days to the starting day (Friday) leads us to Tuesday.

Revision Table: Important Calendar Facts

Concept	Details
Normal Year	365 days
Leap Year	366 days (Feb has 29 days)
Odd Days in a Normal Year	$365 \pmod{7} = 1$
Odd Days in a Leap Year	$366 \pmod{7} = 2$
Odd Days per century (approx)	100 years = 76 normal years + 24 leap years = $76 \times 1 + 24 \times 2 = 76 + 48 = 124$ odd days. $124 \pmod{7} = 5$. (Note: Century years divisible by 400 are leap years)
Odd Days for 100 years	5 odd days
Odd Days for 200 years	$5 \times 2 = 10 \pmod{7} = 3$ odd days
Odd Days for 300 years	$5 \times 3 = 15 \pmod{7} = 1$ odd day
Odd Days for 400 years	$5 \times 4 + 1$ (for the leap century) $= 21 \pmod{7} = 0$ odd days
Day Codes	Sunday=0, Monday=1, Tuesday=2, Wednesday=3, Thursday=4, Friday=5, Saturday=6

Additional Information on Calendar Calculations

Calendar problems often involve calculating the day of the week for a given date or finding the day difference between two dates. Key steps usually include:

Identifying leap years within the period.
Calculating the total number of days between the dates or counting the odd days year by year and then within the remaining period.
Adding the total odd days to the starting day's code and finding the resulting day modulo 7.

Knowing the number of days in each month and which years are leap years is fundamental. Practice with different date ranges will help in mastering this type of problem.

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Saturday

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Sunday

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Monday

List I (Indian States/Union Territories)	List II (No. of Lok Sabha Seats)
(A) NCT of Delhi	(I) 42
(B) Uttar Pradesh	(II) 25
(C) West Bengal	(III) 7
(D) Rajasthan	(IV) 80