In the Gregorian calendar, the last day of a century year (such as 1700, 1800, 1900, 2000) can never fall on which day of the week?

Introduction / Context:
This is a conceptual calendar theory question. Instead of asking for the weekday of a specific date, it asks which weekday is impossible for the last day of any century year. Understanding this requires knowledge of how odd days accumulate over 100 and 400 years in the Gregorian calendar, and what pattern that creates for century endings.

Given Data / Assumptions:

• We consider century years such as 1700, 1800, 1900, 2000 and so on.
• A century year is divisible by 100.
• We use the Gregorian leap year rule: a century year is a leap year only if divisible by 400.
• We focus on the last day of the century, that is 31 December of these years.

Concept / Approach:
In the Gregorian calendar, 400 years form a complete cycle containing exactly 146097 days. When divided by 7, this gives 0 odd days, so the calendar repeats every 400 years. Within this 400 year span, the last days of century years fall only on specific weekdays. Analysing the number of odd days across each 100 year block shows which weekdays are possible and which are not.

Step-by-Step Solution:
Step 1: Over 400 years, there are 97 leap years and 303 normal years, giving 146097 days. Step 2: 146097 divided by 7 leaves remainder 0, so the calendar repeats every 400 years. Step 3: When we compute the day for 31 December of successive century years within a 400 year span, the pattern of weekdays for the last day of centuries turns out to be a limited set. Step 4: Using standard results from calendar theory, the last days of centuries can only be Monday, Wednesday, Friday, or Sunday. Step 5: Therefore, days like Tuesday, Thursday, and Saturday are not possible as the last day of a century year. Step 6: Among the given options Monday, Wednesday, Tuesday, and Friday, the only impossible weekday is Tuesday.

Verification / Alternative check:
You can verify the result for a few well known centuries. For example, 31 December 1900 falls on a Monday, 31 December 2000 falls on a Sunday, and other centuries within the 400 year pattern also yield only Monday, Wednesday, Friday, or Sunday. No standard calendar computation ever produces Tuesday as the last day of a century year, which confirms the theory and the answer.

Why Other Options Are Wrong:
Monday: This is possible and indeed occurs for some centuries such as 1900 in standard references.
Wednesday: This is also a valid weekday for the last day of a century within the 400 year cycle.
Friday: This is likewise a possible last day for certain century years and is therefore not excluded.

Common Pitfalls:
Students sometimes try to memorise the wrong set of impossible days or confuse the last day of a century with the first day of a century. Another common confusion is mixing the Gregorian calendar with other calendar systems. For aptitude exams, you should remember the compact result: the last day of a Gregorian century can only be Monday, Wednesday, Friday, or Sunday, so it can never be Tuesday, Thursday, or Saturday.

Final Answer:
Hence, the last day of a century year cannot be Tuesday.