`{:("Column A"," ", "Column B"),("The count of the numbers between 100 and 300 that are divisible by both 5 and 6",,"The count of the numbers between 100 and 300 that are divisible by either 5 or 6"):}`

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 counts in both Column A and Column B based on the given conditions.

### Step-by-step Solution:

**Step 1: Calculate the count of numbers between 100 and 300 that are divisible by both 5 and 6 (Column A).**

1. **Find the Least Common Multiple (LCM)** of 5 and 6.
- LCM(5, 6) = 30.

2. **Identify the first and last numbers in the range (100 to 300) that are divisible by 30.**
- The first number ≥ 100 that is divisible by 30 is 120.
- The last number ≤ 300 that is divisible by 30 is 300.

3. **Use the formula for the number of terms in an arithmetic sequence:**
\[ T_n = a + (n - 1) \cdot d \]
where:
- \( T_n \) is the last term (300),
- \( a \) is the first term (120),
- \( d \) is the common difference (30).

4. **Set up the equation:**
\[ 300 = 120 + (n - 1) \cdot 30 \]

5. **Solve for \( n \):**
\[ 300 - 120 = (n - 1) \cdot 30 \\ 180 = (n - 1) \cdot 30 \\ n - 1 = \frac{180}{30} \\ n - 1 = 6 \\ n = 7 \]

Thus, the count of numbers between 100 and 300 that are divisible by both 5 and 6 (Column A) is **7**.

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**Step 2: Calculate the count of numbers between 100 and 300 that are divisible by either 5 or 6 (Column B).**

1. **Count the numbers divisible by 5:**
- The first number ≥ 100 that is divisible by 5 is 100.
- The last number ≤ 300 that is divisible by 5 is 300.

2. **Set up the equation for numbers divisible by 5:**
\[ 300 = 100 + (n - 1) \cdot 5 \]

3. **Solve for \( n \):**
\[ 300 - 100 = (n - 1) \cdot 5 \\ 200 = (n - 1) \cdot 5 \\ n - 1 = \frac{200}{5} \\ n - 1 = 40 \\ n = 41 \]

So, there are **41 numbers** between 100 and 300 that are divisible by 5.

4. **Count the numbers divisible by 6:**
- The first number ≥ 100 that is divisible by 6 is 102.
- The last number ≤ 300 that is divisible by 6 is 300.

5. **Set up the equation for numbers divisible by 6:**
\[ 300 = 102 + (n - 1) \cdot 6 \]

6. **Solve for \( n \):**
\[ 300 - 102 = (n - 1) \cdot 6 \\ 198 = (n - 1) \cdot 6 \\ n - 1 = \frac{198}{6} \\ n - 1 = 33 \\ n = 34 \]

So, there are **34 numbers** between 100 and 300 that are divisible by 6.

7. **Use the principle of inclusion-exclusion to find the total count of numbers divisible by either 5 or 6:**
\[ \text{Total} = \text{Count divisible by 5} + \text{Count divisible by 6} - \text{Count divisible by both} \]
\[ \text{Total} = 41 + 34 - 7 = 68 \]

Thus, the count of numbers between 100 and 300 that are divisible by either 5 or 6 (Column B) is **68**.

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### Final Comparison:
- Column A: 7
- Column B: 68

Since 68 > 7, **Column B is larger.**

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