Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.

Options

A. 19668
B. 1966
C. 19999
D. None of these

Explanation

To find the sum of all integers between 100 and 800 that leave a remainder of 7 when divided by 16, we can follow these steps:

### Step 1: Identify the first integer
We need to find the smallest integer greater than 100 that leaves a remainder of 7 when divided by 16.

To do this, we can start by finding the largest multiple of 16 that is less than or equal to 100.
- The largest multiple of 16 less than 100 is 96 (since \(16 \times 6 = 96\)).
- To get a number that leaves a remainder of 7, we add 7 to this multiple:
\[ 96 + 7 = 103 \]
Thus, the first integer is **103**.

### Step 2: Identify the last integer
Next, we need to find the largest integer less than 800 that leaves a remainder of 7 when divided by 16.

The largest multiple of 16 less than or equal to 800 is 800 itself (since \(16 \times 50 = 800\)).
- To find the last integer that leaves a remainder of 7, we subtract 9 from this multiple (since \(16 - 7 = 9\)):
\[ 800 - 9 = 791 \]
Thus, the last integer is **791**.

### Step 3: Form the Arithmetic Progression (AP)
Now we have the first term \(a = 103\) and the last term \(l = 791\). The common difference \(d\) is 16 (since each subsequent term that leaves a remainder of 7 when divided by 16 is obtained by adding 16).

### Step 4: Find the number of terms (n)
The nth term of an AP can be expressed as:
\[ l = a + (n-1)d \]
Substituting the known values:
\[ 791 = 103 + (n-1) \cdot 16 \]
Rearranging gives:
\[ 791 - 103 = (n-1) \cdot 16 \]
\[ 688 = (n-1) \cdot 16 \]
Now, divide both sides by 16:
\[ n - 1 = \frac{688}{16} = 43 \]
Thus,
\[ n = 43 + 1 = 44 \]

### Step 5: Calculate the sum of the AP
The sum \(S_n\) of the first \(n\) terms of an AP can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we have:
\[ S_{44} = \frac{44}{2} \cdot (103 + 791) \]
Calculating the sum inside the parentheses:
\[ 103 + 791 = 894 \]
Now substituting back:
\[ S_{44} = 22 \cdot 894 = 19668 \]

### Final Answer
The sum of all integers between 100 and 800 that leave a remainder of 7 when divided by 16 is **19668**.

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