The average of seven distinct positive integers is 8.Whatis the greatest possible value of one of the intergers ?

Options

A. 9
B. 11
C. 35
D. 45

Answer

To solve the problem, we need to determine the greatest possible value of one of the seven distinct positive integers, given that their average is 8.

### Step-by-Step Solution:

1. **Calculate the Total Sum of the Integers**:
Since the average of the seven integers is 8, we can calculate the total sum of these integers using the formula for average:
\[ \text{Average} = \frac{\text{Sum of integers}}{\text{Number of integers}} \]
Rearranging gives us:
\[ \text{Sum of integers} = \text{Average} \times \text{Number of integers} = 8 \times 7 = 56 \]

**Hint**: Remember that the sum of the integers can be found by multiplying the average by the total number of integers.

2. **Identify the Constraints**:
We know that the integers must be distinct positive integers. This means that all integers must be different and greater than zero.

3. **Maximize One Integer**:
To find the greatest possible value of one integer, we can start by assuming that one of the integers is as large as possible. Let's denote this integer as \( x \).

4. **Calculate the Remaining Sum**:
If \( x \) is the largest integer, the sum of the remaining six integers must be:
\[ 56 - x \]

5. **Find the Minimum Sum of the Remaining Integers**:
The smallest six distinct positive integers are 1, 2, 3, 4, 5, and 6. The sum of these integers is:
\[ 1 + 2 + 3 + 4 + 5 + 6 = 21 \]

6. **Set Up the Inequality**:
For \( x \) to be valid, the remaining sum must be at least 21:
\[ 56 - x \geq 21 \]
Rearranging gives:
\[ x \leq 56 - 21 = 35 \]

7. **Conclusion**:
The maximum value of \( x \) that satisfies this condition is 35. We can check if it is possible to have the integers as 35, 1, 2, 3, 4, and 5:
\[ 35 + 1 + 2 + 3 + 4 + 5 = 50 \]
This does not exceed 56, and all integers are distinct. Therefore, the greatest possible value of one of the integers is:
\[ \boxed{35} \]