When 777777 ............ 363 times, is divided by 11, the remainder obtained is?

Options

A. 10
B. 7
C. 3
D. 8

Answer

To solve the problem of finding the remainder when the number formed by repeating 777 a total of 363 times is divided by 11, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Number**:
The number we are dealing with is 777 repeated 363 times. We can denote this number as \( N = 777777... \) (363 times).

2. **Finding the Remainder of 777**:
First, we need to find the remainder when 777 is divided by 11.
\[ 777 \div 11 = 70 \quad \text{(whole part)} \]
\[ 70 \times 11 = 770 \]
\[ \text{Remainder} = 777 - 770 = 7 \]
So, \( 777 \equiv 7 \mod 11 \).

3. **Finding the Remainder of the Repeated Number**:
Since \( N \) consists of 363 occurrences of 777, we can express \( N \) as:
\[ N = 777 \times (10^{3 \times 362} + 10^{3 \times 361} + ... + 10^0) \]
The sum inside the parentheses is a geometric series with 363 terms, where the first term \( a = 1 \) and the common ratio \( r = 10^3 \).

4. **Calculating the Sum of the Geometric Series**:
The sum \( S \) of the geometric series can be calculated using the formula:
\[ S = \frac{a(r^n - 1)}{r - 1} \]
Here, \( n = 363 \), \( a = 1 \), and \( r = 10^3 \):
\[ S = \frac{1 \times ((10^3)^{363} - 1)}{10^3 - 1} = \frac{(10^{1089} - 1)}{999} \]

5. **Finding the Remainder of \( S \) Modulo 11**:
We need to find \( S \mod 11 \). By Fermat's Little Theorem, since \( 10 \equiv -1 \mod 11 \):
\[ 10^{1089} \equiv (-1)^{1089} \equiv -1 \mod 11 \]
Thus,
\[ 10^{1089} - 1 \equiv -1 - 1 \equiv -2 \mod 11 \]
Now, we need to find \( 999 \mod 11 \):
\[ 999 \div 11 = 90 \quad \text{(whole part)} \]
\[ 90 \times 11 = 990 \]
\[ \text{Remainder} = 999 - 990 = 9 \]
So, \( 999 \equiv 9 \mod 11 \).

6. **Final Calculation**:
Now we can calculate \( S \mod 11 \):
\[ S \equiv \frac{-2}{9} \mod 11 \]
To find \( \frac{-2}{9} \mod 11 \), we need the multiplicative inverse of 9 modulo 11. The inverse of 9 can be found by checking:
\[ 9 \times 5 \equiv 45 \equiv 1 \mod 11 \]
Therefore, the inverse is 5. Now we can compute:
\[ S \equiv -2 \times 5 \equiv -10 \equiv 1 \mod 11 \]

7. **Finding the Final Remainder**:
Now, we multiply this result by the remainder of 777:
\[ N \equiv 7 \times 1 \equiv 7 \mod 11 \]

### Conclusion:
The remainder when the number formed by repeating 777 a total of 363 times is divided by 11 is **7**.