In the month of July 2004, on which dates did Monday occur according to the Gregorian calendar?

Introduction / Context:
This problem asks you to identify all dates in a specific month and year, July 2004, that fall on a Monday. It tests your ability to work with calendars once you know the weekday of one date in the month, using the fact that the pattern of weekdays repeats every 7 days.

Given Data / Assumptions:
- Month and year: July 2004.
- Calendar: Gregorian.
- We must list all dates in July 2004 on which Monday occurs.
- 2004 is a leap year (divisible by 4 and not a century year).

Concept / Approach:
To find all Mondays, you only need to know the weekday for one date in that month. Once you know that, every 7th day from that date is the same weekday. So if, for example, you determine that 5 July 2004 is a Monday, then 12 July, 19 July and 26 July 2004 are also Mondays. Our task is mainly to identify the weekday for the first Monday and then list the sequence.

Step-by-Step Solution:
Step 1: Use a reference date such as 1 January 2004 and compute its weekday, or recall that 1 July 2004 was a Thursday according to standard calendars.Step 2: From 1 July 2004 being Thursday, count forward to find the first Monday in July.Step 3: After Thursday (1st) comes Friday (2nd), Saturday (3rd), Sunday (4th) and then Monday (5th).Step 4: Therefore, 5 July 2004 is the first Monday of the month.Step 5: Now add 7 days repeatedly to get the other Mondays: 5 + 7 = 12, so 12 July 2004 is also a Monday.Step 6: Add another 7: 12 + 7 = 19, so 19 July 2004 is Monday.Step 7: Add another 7: 19 + 7 = 26, so 26 July 2004 is Monday.Step 8: Adding 7 again would give 33, which is not a valid date in July, so we stop.

Verification / Alternative check:
Once you know 5 July 2004 is Monday, you can quickly check the consistency: 6 July Tuesday, 7 July Wednesday, 8 July Thursday, and so on. If you look up any reliable 2004 calendar, it will show Mondays on 5th, 12th, 19th and 26th, confirming your calculation. The regular spacing of 7 days between these dates matches the weekly cycle.

Why Other Options Are Wrong:
Each of the other options lists either dates that do not exist as Mondays in that month or dates that are not all Mondays. For example, 4th, 11th, 18th and 25th are all Sundays in July 2004, not Mondays. The other sets mix dates that fall on different weekdays. Only 5th, 12th, 19th and 26th consistently correspond to Mondays.

Common Pitfalls:
Some students assume that the first day of every month is Monday or that the pattern is the same every year, which is not true. Another common error is miscounting the steps from the known weekday and shifting by one day too many or too few. Carefully track each step and always move in 7 day jumps when listing repeated weekdays.

Final Answer:
In July 2004, Mondays fell on 5th, 12th, 19th and 26th.