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

Options

A. A)7071071
B. B)`7^7`
C. C)7107073
D. 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 first and last four-digit numbers divisible by 7
The smallest four-digit number is 1000. The first four-digit number that is divisible by 7 can be found by dividing 1000 by 7 and rounding up to the nearest whole number, then multiplying by 7.

\[ \text{First four-digit number} = 1001 \quad (\text{since } 1001 \div 7 = 143) \]

The largest four-digit number is 9999. The last four-digit number that is divisible by 7 can be found by dividing 9999 by 7 and rounding down to the nearest whole number, then multiplying by 7.

\[ \text{Last four-digit number} = 9996 \quad (\text{since } 9996 \div 7 = 1428) \]

### Step 2: Determine the number of terms in the series
The series of four-digit numbers divisible by 7 forms an arithmetic sequence where:
- The first term \(a = 1001\)
- The last term \(l = 9996\)
- The common difference \(d = 7\)

To find the number of terms \(n\), we can use the formula for the \(n\)-th term of an arithmetic sequence:

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

Rearranging gives:

\[ n = \frac{l - a}{d} + 1 \]

Substituting the values:

\[ n = \frac{9996 - 1001}{7} + 1 = \frac{8995}{7} + 1 = 1285 + 1 = 1286 \]

### Step 3: Calculate the sum of the series
The sum \(S_n\) of the first \(n\) terms of an arithmetic series can be calculated using the formula:

\[ S_n = \frac{n}{2} \cdot (a + l) \]

Substituting the values we have:

\[ S_{1286} = \frac{1286}{2} \cdot (1001 + 9996) \]

Calculating the sum inside the parentheses:

\[ 1001 + 9996 = 10997 \]

Now substituting back into the sum formula:

\[ S_{1286} = 643 \cdot 10997 \]

Calculating the product:

\[ S_{1286} = 7071071 \]

### Final Answer
The sum of all four-digit numbers which are divisible by 7 is **7071071**.

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