The calendar for the year 2002 is the same as for which of the following year?

Options

A. a) 2006
B. b) 2013
C. c) 2009
D. d) 2010

Answer

To determine which year has the same calendar as the year 2002, we can follow a systematic approach based on the properties of leap years and the day of the week.

### Step-by-Step Solution:

1. **Identify the Year**: We need to find a year that has the same calendar as 2002.

2. **Check if 2002 is a Leap Year**:
- A leap year occurs every 4 years, except for years that are divisible by 100 but not by 400.
- Since 2002 is not divisible by 4, it is not a leap year.

3. **Calculate the Remainder When Dividing by 4**:
- Divide 2002 by 4:
\[ 2002 \div 4 = 500 \quad \text{(remainder 2)} \]

4. **Determine the Year to Add**:
- According to the rule provided, if the remainder is 2, we add 11 to the year:
\[ 2002 + 11 = 2013 \]

5. **Verify the Year**:
- Now we need to check if 2013 is indeed a non-leap year and has the same day of the week for January 1st as 2002.
- 2013 is also not a leap year (not divisible by 4).

6. **Check the Day of the Week**:
- January 1, 2002 was a Tuesday.
- To find the day of the week for January 1, 2013, we can calculate the total number of days between these two years.
- From 2002 to 2013, there are 11 years. In this period, there are 8 non-leap years and 3 leap years (2004, 2008, 2012).
- Total days:
\[ (8 \times 365) + (3 \times 366) = 2920 + 1098 = 4018 \]
- Now, divide 4018 by 7 to find the number of weeks and the remainder:
\[ 4018 \div 7 = 573 \quad \text{(remainder 7)} \]
- Since the remainder is 0, January 1, 2013 is also a Tuesday.

7. **Conclusion**:
- Therefore, the calendar for the year 2002 is the same as the calendar for the year 2013.

### Final Answer:
The calendar for the year 2002 is the same as the year **2013**.

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