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

Options

A. 57
B. 19
C. 38
D. 76

Answer

To solve the problem of finding the number of numbers lying between 100 and 500 that are divisible by 7 but not by 21, we can follow these steps:

### Step 1: Identify the Range
We need to find numbers between 100 and 500.

### Step 2: Find the First and Last Terms Divisible by 7
- The smallest number greater than 100 that is divisible by 7 is 105 (since 105 = 7 × 15).
- The largest number less than 500 that is divisible by 7 is 497 (since 497 = 7 × 71).

### Step 3: Determine the Sequence of Numbers Divisible by 7
The sequence of numbers divisible by 7 between 100 and 500 is:
105, 112, 119, ..., 497.

This is an arithmetic progression (AP) where:
- First term (a) = 105
- Common difference (d) = 7
- Last term (L) = 497

### Step 4: Calculate the Number of Terms (n) in the AP
Using the formula for the nth term of an AP:
\[ L = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 497 = 105 + (n - 1) \cdot 7 \]

Rearranging gives:
\[ (n - 1) \cdot 7 = 497 - 105 \]
\[ (n - 1) \cdot 7 = 392 \]
\[ n - 1 = \frac{392}{7} \]
\[ n - 1 = 56 \]
\[ n = 57 \]

So, there are 57 numbers between 100 and 500 that are divisible by 7.

### Step 5: Find the Numbers Divisible by 21
Next, we need to find the numbers that are divisible by 21 in the same range.
- The smallest number greater than 100 that is divisible by 21 is 105 (since 105 = 21 × 5).
- The largest number less than 500 that is divisible by 21 is 483 (since 483 = 21 × 23).

The sequence of numbers divisible by 21 is:
105, 126, 147, ..., 483.

This is also an AP where:
- First term (a) = 105
- Common difference (d) = 21
- Last term (L) = 483

### Step 6: Calculate the Number of Terms (m) in this AP
Using the same formula for the nth term:
\[ L = a + (m - 1) \cdot d \]
Substituting the known values:
\[ 483 = 105 + (m - 1) \cdot 21 \]

Rearranging gives:
\[ (m - 1) \cdot 21 = 483 - 105 \]
\[ (m - 1) \cdot 21 = 378 \]
\[ m - 1 = \frac{378}{21} \]
\[ m - 1 = 18 \]
\[ m = 19 \]

So, there are 19 numbers between 100 and 500 that are divisible by 21.

### Step 7: Calculate the Numbers Divisible by 7 but Not by 21
To find the numbers that are divisible by 7 but not by 21, we subtract the count of numbers divisible by 21 from those divisible by 7:
\[ \text{Numbers divisible by 7 but not by 21} = n - m \]
\[ = 57 - 19 \]
\[ = 38 \]

### Final Answer
Thus, the number of numbers lying between 100 and 500 that are divisible by 7 but not by 21 is **38**.

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