Find the remainder when (777777 …… 1000 times) is divided by 13.

Options

A. 10
B. 3
C. 4
D. 5

Answer

To find the remainder when the number formed by repeating "777" a total of 1000 times is divided by 13, we can follow these steps:

### Step 1: Understand the structure of the number
The number we are dealing with is "777" repeated 1000 times. This can be expressed as:
\[ N = 777777... \text{ (1000 times)} \]

### Step 2: Break down the number
Since "777" is a 3-digit number, we can express \( N \) as:
\[ N = 777 \times 10^{3 \times (1000 - 1)} + 777 \times 10^{3 \times (1000 - 2)} + ... + 777 \times 10^0 \]

This is a geometric series where the first term \( a = 777 \) and the common ratio \( r = 10^3 = 1000 \).

### Step 3: Calculate the sum of the geometric series
The sum \( S \) of the first \( n \) terms of a geometric series can be calculated using the formula:
\[ S = a \frac{r^n - 1}{r - 1} \]
Here, \( n = 1000 \), \( a = 777 \), and \( r = 1000 \):
\[ S = 777 \frac{1000^{1000} - 1}{1000 - 1} = 777 \frac{1000^{1000} - 1}{999} \]

### Step 4: Find the remainder when divided by 13
Instead of calculating \( S \) directly, we can find the remainder of \( 777 \) and \( 1000^{1000} - 1 \) modulo 13.

#### Step 4.1: Calculate \( 777 \mod 13 \)
First, we find the remainder of \( 777 \) when divided by \( 13 \):
\[ 777 \div 13 = 59 \quad \text{(integer part)} \]
\[ 59 \times 13 = 767 \]
\[ 777 - 767 = 10 \]
So, \( 777 \mod 13 = 10 \).

#### Step 4.2: Calculate \( 1000^{1000} \mod 13 \)
Next, we need to calculate \( 1000 \mod 13 \):
\[ 1000 \div 13 = 76 \quad \text{(integer part)} \]
\[ 76 \times 13 = 988 \]
\[ 1000 - 988 = 12 \]
So, \( 1000 \mod 13 = 12 \).

Now we need to calculate \( 12^{1000} \mod 13 \). By Fermat's Little Theorem, since \( 12 \equiv -1 \mod 13 \):
\[ 12^{12} \equiv 1 \mod 13 \]
Thus, \( 12^{1000} \mod 13 \) can be simplified:
\[ 1000 \mod 12 = 4 \]
So,
\[ 12^{1000} \equiv 12^4 \mod 13 \]

Calculating \( 12^4 \mod 13 \):
\[ 12^2 = 144 \]
\[ 144 \mod 13 = 1 \]
Thus,
\[ 12^4 = (12^2)^2 \equiv 1^2 = 1 \mod 13 \]

#### Step 4.3: Combine results
Now we have:
\[ 1000^{1000} - 1 \equiv 1 - 1 = 0 \mod 13 \]

### Step 5: Final calculation
Now substituting back into our sum:
\[ S \equiv 777 \cdot 0 \mod 13 \]
Thus,
\[ S \equiv 0 \mod 13 \]

### Conclusion
The remainder when \( N \) (the number formed by repeating "777" 1000 times) is divided by 13 is:
\[ \boxed{0} \]