Find the remainder when (22222 ……101 times) is divided by 11.

Options

A. 2
B. 3
C. 4
D. 9

Answer

To find the remainder when the number consisting of 222...2 (with 101 digits of '2' followed by a '1' and then a '0') is divided by 11, we can follow these steps:

### Step 1: Understand the Structure of the Number
The number can be represented as:
\[ N = 222...2 (101 \text{ times}) \]
This means we have 100 digits of '2' and the last digit is '1' followed by a '0'.

### Step 2: Express the Number in a Mathematical Form
The number can be expressed as:
\[ N = 2 \times 10^{100} + 2 \times 10^{99} + 2 \times 10^{98} + \ldots + 2 \times 10^1 + 1 \times 10^0 \]

### Step 3: Simplify the Expression
This can be simplified as:
\[ N = 2(10^{100} + 10^{99} + 10^{98} + \ldots + 10^1) + 1 \]
The sum inside the parentheses is a geometric series.

### Step 4: Calculate the Sum of the Geometric Series
The sum of the geometric series can be calculated using the formula:
\[ S = a \frac{(r^n - 1)}{(r - 1)} \]
where \( a = 10 \), \( r = 10 \), and \( n = 100 \):
\[ S = 10 \frac{(10^{100} - 1)}{(10 - 1)} = \frac{10^{101} - 10}{9} \]

### Step 5: Substitute Back into the Expression for N
Now substituting back into our expression for \( N \):
\[ N = 2 \left( \frac{10^{101} - 10}{9} \right) + 1 \]

### Step 6: Find the Remainder When Divided by 11
To find the remainder of \( N \) when divided by 11, we can use the property of congruences. We will calculate \( 10^k \mod 11 \) for \( k = 0, 1, 2, \ldots, 100 \):
- \( 10^0 \equiv 1 \mod 11 \)
- \( 10^1 \equiv 10 \mod 11 \)
- \( 10^2 \equiv 1 \mod 11 \)
- \( 10^3 \equiv 10 \mod 11 \)
- And so on...

This shows that \( 10^k \mod 11 \) alternates between 1 and 10.

### Step 7: Count the Terms
Since we have 101 terms:
- For even indices (0, 2, 4, ..., 100): there are 51 terms contributing \( 1 \).
- For odd indices (1, 3, 5, ..., 99): there are 50 terms contributing \( 10 \).

### Step 8: Calculate the Total Contribution Modulo 11
The total contribution modulo 11 is:
\[ 51 \times 1 + 50 \times 10 = 51 + 500 = 551 \]

Now, we find \( 551 \mod 11 \):
\[ 551 \div 11 = 50 \quad \text{(remainder 1)} \]

### Step 9: Final Calculation
Thus, the remainder when \( N \) is divided by 11 is:
\[ R = 2 \times (551 \mod 11) + 1 \]
\[ R = 2 \times 1 + 1 = 2 + 1 = 3 \]

### Conclusion
The remainder when the number consisting of 222...2 (101 times) is divided by 11 is **3**.