The sum of the digits of the number `10 ^(n) - 1 ` is 3375. The value of n is

Options

A. 337
B. 375
C. 335
D. 3375

Answer

To find the value of \( n \) such that the sum of the digits of the number \( 10^n - 1 \) equals 3375, we can follow these steps:

### Step 1: Understand the expression \( 10^n - 1 \)
The expression \( 10^n - 1 \) represents a number that consists of \( n \) digits, all of which are 9. For example:
- If \( n = 1 \), \( 10^1 - 1 = 9 \)
- If \( n = 2 \), \( 10^2 - 1 = 99 \)
- If \( n = 3 \), \( 10^3 - 1 = 999 \)

### Step 2: Calculate the sum of the digits
The sum of the digits of \( 10^n - 1 \) can be calculated as follows:
- For \( n = 1 \): The sum of digits is \( 9 \) (which is \( 9 \times 1 \))
- For \( n = 2 \): The sum of digits is \( 18 \) (which is \( 9 \times 2 \))
- For \( n = 3 \): The sum of digits is \( 27 \) (which is \( 9 \times 3 \))

In general, the sum of the digits of \( 10^n - 1 \) is \( 9n \).

### Step 3: Set up the equation
We know from the problem that the sum of the digits is equal to 3375. Therefore, we can set up the equation:
\[ 9n = 3375 \]

### Step 4: Solve for \( n \)
To find \( n \), divide both sides of the equation by 9:
\[ n = \frac{3375}{9} \]

### Step 5: Perform the division
Now, we perform the division:
\[ n = 375 \]

### Conclusion
Thus, the value of \( n \) is \( 375 \).

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