The number of numbers lying between 81 and 1792 which are divisible by 17 is

Options

A. 105
B. 107
C. 109
D. 101

Answer

To find the number of numbers lying between 81 and 1792 that are divisible by 17, we can follow these steps:

### Step 1: Identify the first number greater than 81 that is divisible by 17.
To find the first number greater than 81 that is divisible by 17, we can divide 81 by 17 and round up to the nearest whole number.

\[ \frac{81}{17} \approx 4.7647 \implies \text{Next whole number is } 5 \]

Now, multiply this by 17 to get the first term:

\[ 5 \times 17 = 85 \]

### Step 2: Identify the last number less than 1792 that is divisible by 17.
To find the last number less than 1792 that is divisible by 17, we can divide 1792 by 17 and round down to the nearest whole number.

\[ \frac{1792}{17} \approx 105.4118 \implies \text{Next whole number is } 105 \]

Now, multiply this by 17 to get the last term:

\[ 105 \times 17 = 1785 \]

### Step 3: Determine the number of terms in the arithmetic progression (AP).
The numbers divisible by 17 between 85 and 1785 form an arithmetic progression where:
- First term \(a = 85\)
- Last term \(l = 1785\)
- Common difference \(d = 17\)

The \(n\)th term of an AP can be given by the formula:

\[ l = a + (n-1) \cdot d \]

Substituting the known values:

\[ 1785 = 85 + (n-1) \cdot 17 \]

### Step 4: Solve for \(n\).
Rearranging the equation gives:

\[ 1785 - 85 = (n-1) \cdot 17 \]

\[ 1700 = (n-1) \cdot 17 \]

Now, divide both sides by 17:

\[ n - 1 = \frac{1700}{17} = 100 \]

Adding 1 to both sides gives:

\[ n = 101 \]

### Conclusion
The number of numbers lying between 81 and 1792 that are divisible by 17 is **101**.

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