If the sum of the digits of a number 10 n – 1, where n is a natural number, is equal to 3798, then what is the value of n?
422
Understanding the Problem: Sum of Digits for \(10^n - 1\)
The question asks us to find the value of a natural number \(n\), given that the sum of the digits of the number \(10^n - 1\) is equal to 3798.
To solve this, we first need to understand what the number \(10^n - 1\) looks like for a natural number \(n\).
Exploring the Structure of \(10^n - 1\)
Let's look at a few examples for different values of \(n\):
- If \(n = 1\), \(10^1 - 1 = 10 - 1 = 9\). The number is 9.
- If \(n = 2\), \(10^2 - 1 = 100 - 1 = 99\). The number is 99.
- If \(n = 3\), \(10^3 - 1 = 1000 - 1 = 999\). The number is 999.
- If \(n = 4\), \(10^4 - 1 = 10000 - 1 = 9999\). The number is 9999.
From these examples, we can see a pattern. For any natural number \(n\), the number \(10^n - 1\) is an integer consisting of exactly \(n\) digits, and every single digit is 9.
For example, if \(n = 5\), \(10^5 - 1 = 99999\), which has five digits, all are 9.
Calculating the Sum of Digits
Since the number \(10^n - 1\) is made up of \(n\) digits, and each digit is a 9, the sum of the digits can be easily calculated. The sum of the digits is simply the number of digits multiplied by the value of the digit.
Sum of digits = (Number of digits) \(\times\) (Value of each digit)
In this case, the number of digits is \(n\), and the value of each digit is 9.
So, the sum of the digits of \(10^n - 1\) is \(n \times 9\).
Setting up and Solving the Equation
The problem states that the sum of the digits is 3798. Using our formula for the sum of digits, we can set up an equation:
\(n \times 9 = 3798\)
To find the value of \(n\), we need to divide 3798 by 9.
\(n = \frac{3798}{9}\)
Let's perform the division:
| Step | Calculation | Notes |
|---|---|---|
| 1 | 37 \(\div\) 9 | Gives 4 with remainder 1 |
| 2 | Bring down 9, number is 19. | 19 \(\div\) 9 gives 2 with remainder 1 |
| 3 | Bring down 8, number is 18. | 18 \(\div\) 9 gives 2 with remainder 0 |
Alternatively, we can perform long division:
\( \begin{array}{r} 422 \\ 9 \overline{) 3798} \\ -36 \downarrow \\ \hline 19 \\ -18 \downarrow \\ \hline 18 \\ -18 \\ \hline 0 \end{array} \)
So, the result of the division is 422.
\(n = 422\)
Conclusion
The value of \(n\) that satisfies the condition that the sum of the digits of \(10^n - 1\) is 3798 is 422.
This matches one of the given options.
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| Form of \(10^n - 1\) | A number consisting of \(n\) digits, all of which are 9 (for \(n\) a natural number). |
| Sum of Digits | For \(10^n - 1\), the sum is \(n \times 9\). |
| Problem Equation | \(n \times 9 = 3798\) |
| Solving for \(n\) | \(n = \frac{\text{Sum of Digits}}{9}\) |
Additional Information: Properties of Numbers Like \(10^n - 1\)
Numbers of the form \(10^n - 1\) are closely related to powers of 10. Subtracting 1 from a power of 10 (like 10, 100, 1000, etc.) results in a number consisting entirely of the digit 9.
- \(10^1 - 1 = 9\)
- \(10^2 - 1 = 99\)
- \(10^3 - 1 = 999\)
These numbers are also related to repeating decimals, as \(\frac{1}{9} = 0.111...\), \(\frac{1}{99} = 0.010101...\), and \(\frac{1}{10^n - 1}\) can be related to repeating decimals with a period of length \(n\).
The sum of the digits property used here is a simple and useful trick for this specific form of number, \(10^n - 1\).
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