The set Q consists of 15 numbers whose arithmetic mean is zero? Which of the following must also be zero?
I. The median of the numbers in Q.
II. The mode of the number in Q.
III. The sum of the numbers in Q.

Options

A. I only
B. II only
C. III only
D. I and III only

Answer

To solve the problem, we need to analyze the given information about the set \( Q \) consisting of 15 numbers with an arithmetic mean of zero. We will go through each option to determine which must also be zero.

### Step-by-Step Solution:

1. **Understanding the Arithmetic Mean**:
The arithmetic mean (AM) of a set of numbers is calculated as:
\[ \text{AM} = \frac{\text{Sum of all numbers}}{\text{Total count of numbers}} \]
In this case, the total count of numbers is 15, and we are given that the arithmetic mean is zero.

2. **Setting Up the Equation**:
Since the arithmetic mean is zero, we can set up the equation:
\[ 0 = \frac{\text{Sum of all 15 numbers}}{15} \]
This implies:
\[ \text{Sum of all 15 numbers} = 0 \times 15 = 0 \]

3. **Analyzing the Options**:
- **Option I: The Median of the Numbers in Q**:
The median is the middle value when the numbers are arranged in order. The median does not necessarily have to be zero, as it depends on the specific values of the numbers. Therefore, we cannot conclude that the median must be zero.

- **Option II: The Mode of the Numbers in Q**:
The mode is the number that appears most frequently in the set. Similar to the median, the mode can be any number and does not have to be zero. Thus, we cannot conclude that the mode must be zero.

- **Option III: The Sum of the Numbers in Q**:
From our earlier calculation, we established that the sum of all 15 numbers is indeed zero. Therefore, this option must be true.

4. **Conclusion**:
Based on our analysis:
- The sum of the numbers in \( Q \) must be zero.
- The median and mode do not necessarily have to be zero.

Thus, the correct answer is:
- **Only Option III is correct**.

### Summary of the Results:
- **I. Median**: Not necessarily zero.
- **II. Mode**: Not necessarily zero.
- **III. Sum**: Must be zero.