777777 .......... 129 times is divided by 37, the remainder obtained is ?

Options

A. 0
B. 37
C. 1
D. 36

Answer

To solve the problem of finding the remainder when the number formed by 777... (129 times) is divided by 37, we can follow these steps:

### Step 1: Understand the Number
The number we are dealing with is 777... (129 times). This can be expressed as:
\[ N = 777...777 \text{ (129 times)} \]

### Step 2: Express the Number Mathematically
We can express this number as:
\[ N = 7 \times 10^{128} + 7 \times 10^{127} + 7 \times 10^{126} + \ldots + 7 \times 10^0 \]

This is a geometric series where the first term \( a = 7 \) and the common ratio \( r = 10 \).

### Step 3: Sum the Geometric Series
The sum \( S \) of the first \( n \) terms of a geometric series can be calculated using the formula:
\[ S = a \frac{(r^n - 1)}{(r - 1)} \]

In our case, \( n = 129 \), \( a = 7 \), and \( r = 10 \):
\[ S = 7 \frac{(10^{129} - 1)}{(10 - 1)} \]
\[ S = 7 \frac{(10^{129} - 1)}{9} \]

### Step 4: Find the Remainder when Divided by 37
Now we need to find \( S \mod 37 \):
\[ N = \frac{7(10^{129} - 1)}{9} \]

To find \( N \mod 37 \), we first need to calculate \( 10^{129} \mod 37 \).

### Step 5: Calculate \( 10^{129} \mod 37 \)
Using Fermat's Little Theorem, since 37 is prime:
\[ 10^{36} \equiv 1 \mod 37 \]

Now, we can reduce \( 129 \mod 36 \):
\[ 129 \div 36 = 3 \text{ remainder } 21 \]
So,
\[ 129 \mod 36 = 21 \]
Thus,
\[ 10^{129} \equiv 10^{21} \mod 37 \]

### Step 6: Calculate \( 10^{21} \mod 37 \)
We can calculate \( 10^{21} \mod 37 \) by successive squaring:
- \( 10^1 \equiv 10 \)
- \( 10^2 \equiv 100 \mod 37 \equiv 26 \)
- \( 10^4 \equiv 26^2 \mod 37 \equiv 36 \equiv -1 \)
- \( 10^8 \equiv (-1)^2 \equiv 1 \)
- \( 10^{16} \equiv 1^2 \equiv 1 \)

Now, we can combine these:
\[ 10^{21} = 10^{16} \times 10^4 \times 10^1 \]
\[ 10^{21} \equiv 1 \times (-1) \times 10 \equiv -10 \mod 37 \equiv 27 \]

### Step 7: Substitute Back to Find \( N \mod 37 \)
Now we substitute back:
\[ N \equiv \frac{7(27 - 1)}{9} \mod 37 \]
\[ N \equiv \frac{7 \times 26}{9} \mod 37 \]

Calculating \( 7 \times 26 = 182 \):
Now we need to find \( \frac{182}{9} \mod 37 \).

### Step 8: Find the Modular Inverse of 9 mod 37
We need to find the modular inverse of 9 modulo 37. Using the Extended Euclidean Algorithm, we find that the inverse is 33 (since \( 9 \times 33 \equiv 1 \mod 37 \)).

### Step 9: Calculate the Final Remainder
Now we can calculate:
\[ N \equiv 182 \times 33 \mod 37 \]
Calculating \( 182 \mod 37 \):
\[ 182 \div 37 = 4 \text{ remainder } 34 \]
So,
\[ N \equiv 34 \times 33 \mod 37 \]
Calculating \( 34 \times 33 = 1122 \):
Now find \( 1122 \mod 37 \):
\[ 1122 \div 37 = 30 \text{ remainder } 12 \]

Thus, the remainder when \( 777...777 \) (129 times) is divided by 37 is:
\[ \boxed{12} \]