A list kept at Town Hall contains the twon's average daily temperature in Fahrenheit, rounded to the nearest integer. A particular completed month has either 30 or 31 days. How many days does the month have?
(1) The median temperature is 73.5.
(2) The sum of the average daily temperature is divisible by 3.

Answer

To solve the problem, we need to determine how many days a particular month has based on the given statements regarding the average daily temperature. The month can either have 30 or 31 days, and we have two statements to analyze.

### Step-by-Step Solution:

1. **Understanding the Median Temperature**:
- We are given that the median temperature is 73.5.
- The median is the middle value in a sorted list of numbers. If the number of observations (days in this case) is odd, the median is the middle number. If it is even, the median is the average of the two middle numbers.

2. **Evaluating Statement 1**:
- If the month has **31 days** (an odd number), the median would be the 16th temperature reading in the sorted list. Since the median is given as 73.5, it implies that the 16th reading must be 73.5.
- However, since the average daily temperatures are rounded to the nearest integer, the 16th reading cannot be a decimal (73.5). Therefore, having 31 days contradicts the given median.
- If the month has **30 days** (an even number), the median would be the average of the 15th and 16th readings. In this case, it is possible for the average of two integers to be 73.5 (for example, 73 and 74).
- Thus, Statement 1 alone is sufficient to conclude that the month must have **30 days**.

3. **Evaluating Statement 2**:
- The second statement states that the sum of the average daily temperatures is divisible by 3.
- This condition can be satisfied by both 30 days and 31 days. For instance, if the sum of temperatures for both scenarios is calculated, it can still be divisible by 3 regardless of whether there are 30 or 31 days.
- Therefore, Statement 2 alone is insufficient to determine the number of days in the month.

4. **Conclusion**:
- From the analysis, we conclude that **only Statement 1 is sufficient** to determine that the month has 30 days, while Statement 2 does not provide any conclusive information.

### Final Answer:
The month has **30 days**.