If the sum of ten different positive integers is 100, then what is the greatest possible number among these 10 numbers?

Options

A. 45
B. 91
C. 55
D. 64

Answer

To solve the problem of finding the greatest possible number among ten different positive integers that sum to 100, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find ten different positive integers whose sum is 100. To maximize the largest number, we should minimize the sum of the other nine numbers.

2. **Choosing the Smallest Positive Integers**: The smallest nine different positive integers are 1, 2, 3, 4, 5, 6, 7, 8, and 9.

3. **Calculating the Sum of the Smallest Nine Numbers**:
\[ \text{Sum} = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 \]

4. **Finding the Tenth Number**: Since the total sum must be 100, we can find the tenth number by subtracting the sum of the first nine numbers from 100.
\[ \text{Tenth number} = 100 - 45 = 55 \]

5. **Conclusion**: The greatest possible number among the ten different positive integers is 55.

### Final Answer:
The greatest possible number among these ten different positive integers is **55**.

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