The number of numbers lying between 100 and 500 which are divisible by 7 but not by 21 is

Options

A. 19
B. 38
C. 57
D. none

Answer

To find the number of integers between 100 and 500 that are divisible by 7 but not by 21, we will follow these steps:

### Step 1: Find the numbers between 100 and 500 that are divisible by 7.

1. **Identify the first number greater than 100 that is divisible by 7**:
- The smallest integer greater than 100 that is divisible by 7 can be found by calculating \( 100 \div 7 \) and rounding up to the nearest whole number.
- \( 100 \div 7 \approx 14.2857 \) → Round up to 15.
- Multiply by 7: \( 15 \times 7 = 105 \).

2. **Identify the last number less than 500 that is divisible by 7**:
- The largest integer less than 500 that is divisible by 7 can be found by calculating \( 500 \div 7 \) and rounding down.
- \( 500 \div 7 \approx 71.4286 \) → Round down to 71.
- Multiply by 7: \( 71 \times 7 = 497 \).

3. **Form the arithmetic sequence**:
- The sequence of numbers divisible by 7 between 105 and 497 is \( 105, 112, 119, \ldots, 497 \).
- This is an arithmetic sequence where:
- First term \( a = 105 \)
- Common difference \( d = 7 \)
- Last term \( l = 497 \)

4. **Find the number of terms \( n \) in this sequence**:
- The formula for the nth term of an arithmetic sequence is given by:
\[ l = a + (n-1) \cdot d \]
- Substituting the known values:
\[ 497 = 105 + (n-1) \cdot 7 \]
- Rearranging gives:
\[ 392 = (n-1) \cdot 7 \]
\[ n-1 = \frac{392}{7} = 56 \]
\[ n = 56 + 1 = 57 \]

### Step 2: Find the numbers between 100 and 500 that are divisible by 21.

1. **Identify the first number greater than 100 that is divisible by 21**:
- The smallest integer greater than 100 that is divisible by 21 can be found by calculating \( 100 \div 21 \) and rounding up.
- \( 100 \div 21 \approx 4.7619 \) → Round up to 5.
- Multiply by 21: \( 5 \times 21 = 105 \).

2. **Identify the last number less than 500 that is divisible by 21**:
- The largest integer less than 500 that is divisible by 21 can be found by calculating \( 500 \div 21 \) and rounding down.
- \( 500 \div 21 \approx 23.8095 \) → Round down to 23.
- Multiply by 21: \( 23 \times 21 = 483 \).

3. **Form the arithmetic sequence**:
- The sequence of numbers divisible by 21 between 105 and 483 is \( 105, 126, 147, \ldots, 483 \).
- This is an arithmetic sequence where:
- First term \( a = 105 \)
- Common difference \( d = 21 \)
- Last term \( l = 483 \)

4. **Find the number of terms \( m \) in this sequence**:
- Using the nth term formula:
\[ 483 = 105 + (m-1) \cdot 21 \]
- Rearranging gives:
\[ 378 = (m-1) \cdot 21 \]
\[ m-1 = \frac{378}{21} = 18 \]
\[ m = 18 + 1 = 19 \]

### Step 3: Calculate the required numbers.

1. **Subtract the count of numbers divisible by 21 from those divisible by 7**:
- The total numbers divisible by 7 is \( n = 57 \).
- The total numbers divisible by 21 is \( m = 19 \).
- Therefore, the required numbers are:
\[ \text{Required numbers} = n - m = 57 - 19 = 38 \]

### Final Answer:
The number of integers between 100 and 500 that are divisible by 7 but not by 21 is **38**.

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