The mean of five positive integers is 5. The numbers have a single mode equal to 8 . What is the maximum possible value of the lowest term ?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we need to find the maximum possible value of the lowest term among five positive integers whose mean is 5 and which have a single mode equal to 8.

### Step 1: Understand the Mean
The mean of five positive integers is given as 5. Therefore, we can express this mathematically:
\[ \frac{a + b + c + d + e}{5} = 5 \]
Multiplying both sides by 5 gives us:
\[ a + b + c + d + e = 25 \]

### Step 2: Identify the Mode
The problem states that the numbers have a single mode equal to 8. This means that the number 8 must appear more frequently than any other number in the set. For a number to be the mode, it must appear at least twice.

### Step 3: Determine the Frequency of 8
Let’s consider how many times the number 8 can appear:
- **Case 1**: 8 appears twice.
- **Case 2**: 8 appears three times.
- **Case 3**: 8 appears four times.
- **Case 4**: 8 appears five times.

**Case 1**: If 8 appears twice, we can denote it as:
\[ d = 8, \quad e = 8 \]
Thus, we have:
\[ a + b + c + 8 + 8 = 25 \implies a + b + c = 9 \]

**Case 2**: If 8 appears three times, we denote it as:
\[ c = 8, \quad d = 8, \quad e = 8 \]
Thus, we have:
\[ a + b + 8 + 8 + 8 = 25 \implies a + b = 1 \]
This case is not possible since both \(a\) and \(b\) must be positive integers.

**Case 3**: If 8 appears four times, we denote it as:
\[ b = 8, \quad c = 8, \quad d = 8, \quad e = 8 \]
Thus, we have:
\[ a + 8 + 8 + 8 + 8 = 25 \implies a = 1 \]
This case is also not valid for the same reason as Case 2.

**Case 4**: If 8 appears five times, it contradicts the condition of having a single mode.

### Step 4: Valid Case
The only valid case is Case 1, where 8 appears twice. We have:
\[ a + b + c = 9 \]
Now, we need to find combinations of \(a\), \(b\), and \(c\) such that they are positive integers and the lowest term is maximized.

### Step 5: Finding Combinations
We need to find positive integers \(a\), \(b\), and \(c\) that sum to 9. The combinations could be:
- \(1, 1, 7\)
- \(1, 2, 6\)
- \(1, 3, 5\)
- \(2, 2, 5\)
- \(2, 3, 4\)

However, we need to ensure that the mode remains 8.

### Step 6: Maximizing the Lowest Term
To maximize the lowest term, we should try to keep the smallest number as large as possible.

- If we take \(1, 2, 6\), the lowest term is 1.
- If we take \(2, 2, 5\), the lowest term is 2.
- If we take \(2, 3, 4\), the lowest term is 2.

The maximum possible value of the lowest term is therefore 2.

### Conclusion
Thus, the maximum possible value of the lowest term is:
\[ \boxed{2} \]