`{:("Column A","The sum of the positive integers from 1 through n can be calculated by the formula"(n(n+1))/(2) ,"Column B"),("The sum of the multiples of 6 between 0 and 100", ,"The sum of the multiples of 8 between 0 and 100"):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to calculate the sums of the multiples of 6 and 8 between 0 and 100, and then compare the two sums.

### Step 1: Calculate the sum of multiples of 6 between 0 and 100.

1. **Identify the first and last terms**:
- The first multiple of 6 is 6.
- The last multiple of 6 less than or equal to 100 is 96.

2. **Determine the number of terms (n)**:
- We can use the formula for the nth term of an arithmetic sequence:
\[ A_n = A_1 + (n-1) \cdot D \]
where \(A_n\) is the last term (96), \(A_1\) is the first term (6), and \(D\) is the common difference (6).
- Setting up the equation:
\[ 96 = 6 + (n-1) \cdot 6 \]
- Rearranging gives:
\[ 96 - 6 = (n-1) \cdot 6 \implies 90 = (n-1) \cdot 6 \]
- Dividing both sides by 6:
\[ n - 1 = 15 \implies n = 16 \]

3. **Calculate the sum of the first n terms**:
- The sum \(S_n\) of the first n terms can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (A_1 + A_n) \]
- Substituting the values:
\[ S_{16} = \frac{16}{2} \cdot (6 + 96) = 8 \cdot 102 = 816 \]

### Step 2: Calculate the sum of multiples of 8 between 0 and 100.

1. **Identify the first and last terms**:
- The first multiple of 8 is 8.
- The last multiple of 8 less than or equal to 100 is also 96.

2. **Determine the number of terms (n)**:
- Using the same formula:
\[ A_n = A_1 + (n-1) \cdot D \]
- Setting up the equation:
\[ 96 = 8 + (n-1) \cdot 8 \]
- Rearranging gives:
\[ 96 - 8 = (n-1) \cdot 8 \implies 88 = (n-1) \cdot 8 \]
- Dividing both sides by 8:
\[ n - 1 = 11 \implies n = 12 \]

3. **Calculate the sum of the first n terms**:
- Using the sum formula:
\[ S_n = \frac{n}{2} \cdot (A_1 + A_n) \]
- Substituting the values:
\[ S_{12} = \frac{12}{2} \cdot (8 + 96) = 6 \cdot 104 = 624 \]

### Step 3: Compare the sums from Column A and Column B.

- **Sum of multiples of 6 (Column A)**: 816
- **Sum of multiples of 8 (Column B)**: 624

Since \(816 > 624\), we conclude that Column A is larger.

### Final Answer:
Column A is larger than Column B.

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