When number `10^1 + 10^2 + 10^3 + 10^4 + ............. + 10^11` is divided by 6, the remain- der will be?

Options

A. 4
B. 2
C. 5
D. 1

Answer

To solve the problem of finding the remainder when the sum \(10^1 + 10^2 + 10^3 + 10^4 + \ldots + 10^{11}\) is divided by 6, we can follow these steps:

### Step 1: Identify the pattern of \(10^n \mod 6\)

First, we need to find the remainder of \(10^n\) when divided by 6 for different values of \(n\).

- For \(n = 1\):
\[ 10^1 = 10 \quad \Rightarrow \quad 10 \mod 6 = 4 \]

- For \(n = 2\):
\[ 10^2 = 100 \quad \Rightarrow \quad 100 \mod 6 = 4 \]

- For \(n = 3\):
\[ 10^3 = 1000 \quad \Rightarrow \quad 1000 \mod 6 = 4 \]

Continuing this pattern, we can see that:
\[ 10^n \mod 6 = 4 \quad \text{for all } n \geq 1 \]

### Step 2: Calculate the total sum of the powers

Now, we can sum the remainders for each term from \(10^1\) to \(10^{11}\):
\[ 10^1 + 10^2 + 10^3 + \ldots + 10^{11} \equiv 4 + 4 + 4 + \ldots + 4 \quad (\text{11 times}) \]

### Step 3: Compute the total remainder

The total sum can be calculated as:
\[ \text{Total} = 4 \times 11 = 44 \]

### Step 4: Find the remainder when the total is divided by 6

Now we need to find \(44 \mod 6\):
\[ 44 \div 6 = 7 \quad \text{(which gives a quotient of 7 and a remainder)} \]
Calculating the remainder:
\[ 44 - (6 \times 7) = 44 - 42 = 2 \]

### Final Answer

Thus, the remainder when \(10^1 + 10^2 + 10^3 + \ldots + 10^{11}\) is divided by 6 is:
\[ \boxed{2} \]