The sum of all four digit numbers which are divisible by 7 is :

Options

A. 7071071
B. `7^7`
C. `7107073`
D. `10019996`

Answer

To find the sum of all four-digit numbers that are divisible by 7, we can follow these steps:

### Step 1: Identify the smallest four-digit number divisible by 7
The smallest four-digit number is 1000. We need to find the smallest four-digit number that is divisible by 7.

1. Divide 1000 by 7:
\[ 1000 \div 7 = 142 \quad \text{(remainder 6)} \]
This means that 1000 is not divisible by 7.

2. To find the smallest four-digit number divisible by 7, we subtract the remainder from 1000:
\[ 1000 - 6 = 994 \quad \text{(not a four-digit number)} \]
So, we add 7 to 994:
\[ 994 + 7 = 1001 \]
Thus, the smallest four-digit number divisible by 7 is **1001**.

### Step 2: Identify the largest four-digit number divisible by 7
The largest four-digit number is 9999. We need to find the largest four-digit number that is divisible by 7.

1. Divide 9999 by 7:
\[ 9999 \div 7 = 1428 \quad \text{(remainder 3)} \]
This means that 9999 is not divisible by 7.

2. To find the largest four-digit number divisible by 7, we subtract the remainder from 9999:
\[ 9999 - 3 = 9996 \]
Thus, the largest four-digit number divisible by 7 is **9996**.

### Step 3: Determine the number of terms in the sequence
The sequence of four-digit numbers divisible by 7 forms an arithmetic progression (AP) where:
- First term \(a = 1001\)
- Last term \(l = 9996\)
- Common difference \(d = 7\)

To find the number of terms \(n\), we can use the formula for the \(n\)th term of an AP:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 9996 = 1001 + (n - 1) \cdot 7 \]
Rearranging gives:
\[ 9996 - 1001 = (n - 1) \cdot 7 \]
\[ 8995 = (n - 1) \cdot 7 \]
\[ n - 1 = \frac{8995}{7} \]
Calculating:
\[ n - 1 = 1285 \quad \Rightarrow \quad n = 1286 \]

### Step 4: Calculate the sum of the arithmetic series
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 found:
\[ S_{1286} = \frac{1286}{2} \cdot (1001 + 9996) \]
Calculating:
\[ S_{1286} = 643 \cdot 10997 \]
Now, performing the multiplication:
\[ S_{1286} = 7071071 \]

### Conclusion
The sum of all four-digit numbers that are divisible by 7 is **7071071**.

---