Reasoning — Calendar

A normal year has 365 days = 52 weeks + 1 odd day. A leap year has 366 days = 52 weeks + 2 odd days. In 100 years, there are 76 normal years and 24 leap years (100 is not a leap year, so 25 - 1 = 24 leap years). Number of odd days = (76 * 1) + (24 * 2) = 76 + 48 = 124 days. To find odd days, divide by 7: 124 / 7 = 17 weeks + 5 odd days. So, there are 5 odd days in 100 years.

The year 2023 is a normal year (not a leap year). The year 2024 is a leap year, and the period from 15th August 2023 to 15th August 2024 includes the leap day (29th February 2024). Number of odd days between 15th August 2023 and 15th August 2024 = 2 (due to crossing the leap day). If 15th August 2023 was Tuesday, then 15th August 2024 will be Tuesday + 2 days = Thursday.

A year is a leap year if it is divisible by 4. However, if a year is a century year (ends in 00), it must be divisible by 400 to be a leap year. 1) 1990 is not divisible by 4. 2) 2002 is not divisible by 4. 3) 2000 is a century year. It is divisible by 400 (2000 / 400 = 5). So, 2000 is a leap year. 4) 2010 is not divisible by 4. Therefore, 2000 is the leap year.

To find the day of the week, we calculate the total number of odd days up to the given date. 1. **Odd days in 1600 years:** 1600 = 4 * 400, so 0 odd days. 2. **Odd days in 300 years (1601-1900):** 300 years have 1 odd day. 3. **Odd days in 46 years (1901-1946):** Number of leap years in 46 years = 46 / 4 = 11 (1904, 1908, ..., 1944). Number of normal years = 46 - 11 = 35. Odd days = (11 * 2) + (35 * 1) = 22 + 35 = 57 days. 57 / 7 = 8 weeks + 1 odd day. 4. **Odd days in 1947 (up to 15th August):** January: 3 (31%7) February: 0 (28%7, 1947 is not a leap year) March: 3 (31%7) April: 2 (30%7) May: 3 (31%7) June: 2 (30%7) July: 3 (31%7) August: 15 days = 15%7 = 1 odd day. Total odd days in 1947 = 3 + 0 + 3 + 2 + 3 + 2 + 3 + 1 = 17 days. 17 / 7 = 2 weeks + 3 odd days. 5. **Total odd days:** 0 (1600) + 1 (300) + 1 (46 years) + 3 (1947) = 5 odd days. Mapping of odd days to days of the week: 0 = Sunday, 1 = Monday, 2 = Tuesday, 3 = Wednesday, 4 = Thursday, 5 = Friday, 6 = Saturday. Since there are 5 odd days, the day was Friday.

We need to find the number of odd days between 10th March 2023 and 10th March 2027. * From 10th March 2023 to 10th March 2024: 2024 is a leap year, and February 29th, 2024 falls within this period. So, 2 odd days. * From 10th March 2024 to 10th March 2025: 2025 is a normal year. So, 1 odd day. * From 10th March 2025 to 10th March 2026: 2026 is a normal year. So, 1 odd day. * From 10th March 2026 to 10th March 2027: 2027 is a normal year. So, 1 odd day. Total odd days = 2 + 1 + 1 + 1 = 5 odd days. If 10th March 2023 was Friday, then 10th March 2027 will be Friday + 5 days. Friday + 5 days = Wednesday.

A calendar repeats when the total number of odd days between the two years is 0 (or a multiple of 7) and both years are of the same type (normal or leap). We calculate the cumulative odd days starting from 2011: * 2011: 1 odd day (normal year) * 2012: 2 odd days (leap year) * 2013: 1 odd day (normal year) * 2014: 1 odd day (normal year) * 2015: 1 odd day (normal year) * 2016: 2 odd days (leap year) * 2017: 1 odd day (normal year) * 2018: 1 odd day (normal year) * 2019: 1 odd day (normal year) * 2020: 2 odd days (leap year) * 2021: 1 odd day (normal year) Cumulative odd days: 2011: 1 2012: 1+2 = 3 2013: 3+1 = 4 2014: 4+1 = 5 2015: 5+1 = 6 2016: 6+2 = 8 (8%7 = 1) 2017: 1+1 = 2 2018: 2+1 = 3 2019: 3+1 = 4 2020: 4+2 = 6 2021: 6+1 = 7 (7%7 = 0) The total number of odd days becomes 0 (or a multiple of 7) at the end of 2021. Since 2011 is a normal year and 2022 is also a normal year, the calendar for 2022 will be the same as 2011.

We are moving backward from 18th February 2005 to 18th February 2004. The year 2004 was a leap year. When moving from a date in year Y to the same date in year Y-1, we usually subtract 1 day for a normal year. However, if the period includes February 29th of the earlier year (Y-1), we subtract 2 days. Here, from 18th February 2005 to 18th February 2004, we cross February 29th, 2004. So, we subtract 2 days from Friday. Friday - 2 days = Wednesday.

We need to calculate the number of odd days between 1st April 2023 and 1st July 2023. * April (30 days): 30 % 7 = 2 odd days. * May (31 days): 31 % 7 = 3 odd days. * June (30 days): 30 % 7 = 2 odd days. * July: We are looking for 1st July, so we don't count days in July. Total odd days = 2 (April) + 3 (May) + 2 (June) = 7 odd days. 7 % 7 = 0 odd days. Since the total number of odd days is 0, the day of the week will be the same. If 1st April 2023 was a Saturday, then 1st July 2023 will also be a Saturday.

We need to find the day of the week for 29th February 2000, given that 29th February 2004 was a Tuesday. We are moving backward in time. Number of odd days between 29th February 2000 and 29th February 2004: * From 29 Feb 2000 to 29 Feb 2001: 2000 is a leap year, so 366 days = 2 odd days. * From 29 Feb 2001 to 29 Feb 2002: 2001 is a normal year, so 365 days = 1 odd day. * From 29 Feb 2002 to 29 Feb 2003: 2002 is a normal year, so 365 days = 1 odd day. * From 29 Feb 2003 to 29 Feb 2004: 2003 is a normal year, so 365 days = 1 odd day. Total odd days from 29 Feb 2000 to 29 Feb 2004 = 2 + 1 + 1 + 1 = 5 odd days. This means 29th February 2004 was 5 days *after* 29th February 2000. So, 29th February 2000 = 29th February 2004 - 5 days. Given 29th February 2004 was Tuesday. Tuesday - 5 days = Tuesday + 2 days = Thursday.

The last day of a century is determined by the number of odd days in that century. We know the number of odd days in: * 100 years = 5 odd days (Friday) * 200 years = 5 * 2 = 10 % 7 = 3 odd days (Wednesday) * 300 years = 5 * 3 = 15 % 7 = 1 odd day (Monday) * 400 years = 5 * 4 + 1 (for the leap century year 400) = 21 % 7 = 0 odd days (Sunday) So, the last day of a century can be Friday, Wednesday, Monday, or Sunday. The days that cannot be the last day of a century are Tuesday, Thursday, and Saturday. From the given options, Thursday cannot be the last day of a century.