When `10^1+10^2+10^3+.......+10^1000+10^1001` is divided by 6, the remainder obtained is ?

Options

A. 4
B. 6
C. 2
D. 3

Answer

To solve the problem of finding the remainder when \(10^1 + 10^2 + 10^3 + \ldots + 10^{1000} + 10^{1001}\) is divided by 6, we can follow these steps:

### Step 1: Understand the expression
The expression we need to evaluate is:
\[ S = 10^1 + 10^2 + 10^3 + \ldots + 10^{1001} \]

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

Calculating \(10 \mod 6\):
\[ 10 \div 6 = 1 \quad \text{(remainder 4)} \]
Thus,
\[ 10 \equiv 4 \mod 6 \]

Now, we can find \(10^n \mod 6\):
\[ 10^1 \equiv 4 \mod 6 \]
\[ 10^2 \equiv 4^2 = 16 \equiv 4 \mod 6 \]
\[ 10^3 \equiv 4^3 = 64 \equiv 4 \mod 6 \]
Continuing this pattern, we see that:
\[ 10^n \equiv 4 \mod 6 \quad \text{for all } n \geq 1 \]

### Step 3: Count the number of terms
The series \(10^1 + 10^2 + 10^3 + \ldots + 10^{1001}\) has a total of:
\[ 1001 \text{ terms} \]

### Step 4: Calculate the sum modulo 6
Since each term \(10^n\) contributes a remainder of 4 when divided by 6, the total contribution of all terms is:
\[ S \equiv 4 + 4 + 4 + \ldots + 4 \quad (1001 \text{ times}) \]
This can be expressed as:
\[ S \equiv 1001 \times 4 \mod 6 \]

### Step 5: Calculate \(1001 \times 4\)
Calculating \(1001 \times 4\):
\[ 1001 \times 4 = 4004 \]

### Step 6: Find the remainder of 4004 when divided by 6
Now we need to find \(4004 \mod 6\):
\[ 4004 \div 6 = 667 \quad \text{(remainder 2)} \]
Thus,
\[ 4004 \equiv 2 \mod 6 \]

### Final Answer
The remainder when \(10^1 + 10^2 + 10^3 + \ldots + 10^{1001}\) is divided by 6 is:
\[ \boxed{2} \]