When `10^1 + 10^2 + 10^3 + ........ + 10^99 + 10^100` is divided by 6, the remainder obtained is

Options

A. 2
B. 4
C. 0
D. 1

Answer

To solve the problem of finding the remainder when \(10^1 + 10^2 + 10^3 + \ldots + 10^{100}\) is divided by 6, we can follow these steps:

### Step 1: Understand the expression
We need to evaluate the sum:
\[ S = 10^1 + 10^2 + 10^3 + \ldots + 10^{100} \]

### Step 2: Find the remainder of \(10^n\) when divided by 6
First, we will find the remainder of \(10^n\) for any \(n\) when divided by 6.

Calculating the first few powers:
- \(10^1 = 10\) and \(10 \mod 6 = 4\)
- \(10^2 = 100\) and \(100 \mod 6 = 4\)
- \(10^3 = 1000\) and \(1000 \mod 6 = 4\)

We can observe that:
\[ 10^n \mod 6 = 4 \quad \text{for all } n \geq 1 \]

### Step 3: Count the number of terms
The series \(10^1 + 10^2 + 10^3 + \ldots + 10^{100}\) has 100 terms.

### Step 4: Calculate the total remainder
Since each term \(10^n\) contributes a remainder of 4, the total contribution from all 100 terms is:
\[ 100 \times 4 = 400 \]

### Step 5: Find the remainder of the total sum when divided by 6
Now, we need to find the remainder of 400 when divided by 6:
\[ 400 \div 6 = 66 \quad \text{(quotient)} \]
Calculating the product:
\[ 66 \times 6 = 396 \]
Now, subtract this from 400 to find the remainder:
\[ 400 - 396 = 4 \]

### Conclusion
Thus, the remainder when \(10^1 + 10^2 + 10^3 + \ldots + 10^{100}\) is divided by 6 is:
\[ \boxed{4} \]