Find the remainder when (19191919 …….. 8 times) is divided by 7.

Options

A. 1
B. 2
C. 4
D. 6

Answer

To find the remainder when the number formed by repeating "19" eight times (i.e., 19191919... for a total of 8 times) is divided by 7, we can follow these steps:

### Step 1: Understand the number
The number can be represented as:
\[ N = 19191919... \text{ (8 times)} \]
This can be expressed as:
\[ N = 19 \times 10^{14} + 19 \times 10^{12} + 19 \times 10^{10} + 19 \times 10^8 + 19 \times 10^6 + 19 \times 10^4 + 19 \times 10^2 + 19 \]

### Step 2: Factor out 19
We can factor out 19 from the expression:
\[ N = 19 \times (10^{14} + 10^{12} + 10^{10} + 10^8 + 10^6 + 10^4 + 10^2 + 1) \]

### Step 3: Simplify the expression inside the parentheses
The expression inside the parentheses is a geometric series. The sum of a geometric series can be calculated using the formula:
\[ S = a \frac{r^n - 1}{r - 1} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

In our case:
- \( a = 1 \)
- \( r = 10^2 = 100 \)
- \( n = 8 \)

So, we have:
\[ S = 1 \frac{100^8 - 1}{100 - 1} = \frac{100^8 - 1}{99} \]

### Step 4: Calculate \( N \mod 7 \)
Now we need to find \( N \mod 7 \):
\[ N \equiv 19 \times S \mod 7 \]

First, we calculate \( 19 \mod 7 \):
\[ 19 \div 7 = 2 \quad \text{(remainder 5)} \]
So, \( 19 \equiv 5 \mod 7 \).

Next, we need to find \( S \mod 7 \):
We can calculate \( 100 \mod 7 \):
\[ 100 \div 7 = 14 \quad \text{(remainder 2)} \]
So, \( 100 \equiv 2 \mod 7 \).

Now we need to calculate \( 100^8 \mod 7 \):
Using Fermat's Little Theorem, since 7 is prime:
\[ 2^{6} \equiv 1 \mod 7 \]
Thus:
\[ 100^8 \equiv 2^8 \mod 7 \]
Calculating \( 2^8 \mod 7 \):
\[ 2^8 = 256 \]
Now, \( 256 \div 7 = 36 \quad \text{(remainder 4)} \)
So, \( 2^8 \equiv 4 \mod 7 \).

Now, we can find \( S \mod 7 \):
\[ S = \frac{100^8 - 1}{99} \equiv \frac{4 - 1}{99} \mod 7 \]
Calculating \( 4 - 1 = 3 \), we need \( 99 \mod 7 \):
\[ 99 \div 7 = 14 \quad \text{(remainder 0)} \]
So, \( 99 \equiv 0 \mod 7 \).

This means we cannot directly compute \( S \) as \( 99 \) is divisible by \( 7 \). However, we can find \( S \) using the terms directly.

### Step 5: Calculate \( S \) directly
Now, we can calculate \( S \) directly:
\[ S = 10^{14} + 10^{12} + 10^{10} + 10^8 + 10^6 + 10^4 + 10^2 + 1 \]
Calculating each term modulo \( 7 \):
- \( 10^1 \equiv 3 \mod 7 \)
- \( 10^2 \equiv 2 \mod 7 \)
- \( 10^3 \equiv 6 \mod 7 \)
- \( 10^4 \equiv 4 \mod 7 \)
- \( 10^5 \equiv 5 \mod 7 \)
- \( 10^6 \equiv 1 \mod 7 \)

Thus, we can see that \( 10^{n} \mod 7 \) will repeat every 6 terms. Therefore:
- \( 10^{14} \equiv 3 \)
- \( 10^{12} \equiv 2 \)
- \( 10^{10} \equiv 6 \)
- \( 10^8 \equiv 4 \)
- \( 10^6 \equiv 1 \)
- \( 10^4 \equiv 5 \)
- \( 10^2 \equiv 2 \)
- \( 1 \equiv 1 \)

Adding these up:
\[ S \equiv 3 + 2 + 6 + 4 + 1 + 5 + 2 + 1 \mod 7 \]
\[ S \equiv 24 \mod 7 \]
Calculating \( 24 \mod 7 \):
\[ 24 \div 7 = 3 \quad \text{(remainder 3)} \]
So, \( S \equiv 3 \mod 7 \).

### Step 6: Final calculation for \( N \)
Now we can find \( N \mod 7 \):
\[ N \equiv 5 \times 3 \mod 7 \]
Calculating:
\[ 5 \times 3 = 15 \]
Now, \( 15 \mod 7 \):
\[ 15 \div 7 = 2 \quad \text{(remainder 1)} \]
So, \( N \equiv 1 \mod 7 \).

### Conclusion
The remainder when \( 19191919... \) (8 times) is divided by 7 is:
\[ \boxed{1} \]