A certain number N when multiplied by 13, the resultant values consists entirely of sevens. The value of N is :

Options

A. 123459
B. 58829
C. 59829
D. None of these

Answer

To find the value of the number \( N \) such that when multiplied by 13, the result consists entirely of the digit 7, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( N \) such that:
\[ 13 \times N = 777...7 \]
where the right side consists entirely of the digit 7.

### Step 2: Express the Result in Terms of \( N \)
The number consisting entirely of sevens can be expressed as:
\[ 777...7 = 7 \times \frac{10^k - 1}{9} \]
where \( k \) is the number of sevens. This is because \( 777...7 \) can be seen as \( 7 \) repeated \( k \) times.

### Step 3: Set Up the Equation
We can set up the equation:
\[ 13N = 7 \times \frac{10^k - 1}{9} \]

### Step 4: Solve for \( N \)
Rearranging the equation gives:
\[ N = \frac{7 \times (10^k - 1)}{117} \]
since \( 13 \times 9 = 117 \).

### Step 5: Find Suitable \( k \)
To find \( N \), we need \( 10^k - 1 \) to be divisible by 117. We can check values of \( k \) to find the smallest \( N \).

### Step 6: Check Values of \( k \)
Let's check \( k = 3 \):
- \( 10^3 - 1 = 999 \)
- \( 999 \div 117 = 8.54 \) (not an integer)

Now check \( k = 6 \):
- \( 10^6 - 1 = 999999 \)
- \( 999999 \div 117 = 8547 \) (an integer)

### Step 7: Calculate \( N \)
Now substitute \( k = 6 \) back into the equation for \( N \):
\[ N = \frac{7 \times 999999}{117} \]
Calculating this gives:
\[ N = \frac{6999993}{117} = 59999 \]

### Step 8: Conclusion
Thus, the value of \( N \) is:
\[ N = 59999 \]