`(10^(10)+10^(100)+10^(1000)+ -----+10^10000000000)/(7)` find R.

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem \((10^{10} + 10^{100} + 10^{1000} + \ldots + 10^{10000000000}) / 7\) and find the remainder \(R\), we will follow these steps:

### Step 1: Identify the Pattern in Remainders
First, we need to find the pattern of remainders when powers of 10 are divided by 7. We can calculate the first few powers of 10 modulo 7:

- \(10^1 \mod 7 = 3\)
- \(10^2 \mod 7 = 2\)
- \(10^3 \mod 7 = 6\)
- \(10^4 \mod 7 = 4\)
- \(10^5 \mod 7 = 5\)
- \(10^6 \mod 7 = 1\)

After \(10^6\), the remainders repeat every 6 terms. Thus, the cycle of remainders is: \(3, 2, 6, 4, 5, 1\).

**Hint:** Calculate the remainders of \(10^n\) for \(n = 1\) to \(6\) to find the repeating pattern.

### Step 2: Determine the Exponents Modulo 6
Next, we need to determine the exponent of each term in the series modulo 6 to find the corresponding remainder:

- For \(10^{10}\): \(10 \mod 6 = 4\) (so remainder is \(10^4 \mod 7 = 4\))
- For \(10^{100}\): \(100 \mod 6 = 4\) (so remainder is \(10^4 \mod 7 = 4\))
- For \(10^{1000}\): \(1000 \mod 6 = 4\) (so remainder is \(10^4 \mod 7 = 4\))
- Continuing this way, all terms \(10^{10^n}\) for \(n \geq 1\) will also yield \(10^4 \mod 7 = 4\).

**Hint:** Use \(n \mod 6\) to find the corresponding remainder for each exponent.

### Step 3: Count the Number of Terms
The series has terms of the form \(10^{10^n}\) where \(n\) ranges from 1 to 10. Thus, there are 10 terms in total.

**Hint:** Count the number of terms in the series to ensure you include all contributions.

### Step 4: Sum the Remainders
Now, we can sum the remainders of all terms:

- Each term contributes a remainder of \(4\).
- Total contribution from all 10 terms: \(10 \times 4 = 40\).

**Hint:** Multiply the number of terms by the common remainder to find the total sum of remainders.

### Step 5: Find the Final Remainder
Finally, we need to find \(40 \mod 7\):

\[ 40 \div 7 = 5 \quad \text{(with a remainder of 5)} \]

Thus, \(40 \mod 7 = 5\).

### Conclusion
The remainder \(R\) when \((10^{10} + 10^{100} + 10^{1000} + \ldots + 10^{10000000000})\) is divided by \(7\) is:

\[ \boxed{5} \]