What will be the value of x for `((100^(17)-1)+(10^(34)+x))/(9)`, the remainder = 0

Options

A. 3
B. 6
C. 9
D. 8

Answer

To solve the problem, we need to find the value of \( x \) such that

\[ \frac{(100^{17} - 1) + (10^{34} + x)}{9} \]

gives a remainder of 0. This means that the expression

\[ (100^{17} - 1) + (10^{34} + x) \]

must be divisible by 9.

### Step-by-Step Solution:

1. **Calculate \( 100^{17} \mod 9 \)**:
- First, we simplify \( 100 \mod 9 \):
\[ 100 \div 9 = 11 \quad \text{(remainder 1)} \]
Thus, \( 100 \equiv 1 \mod 9 \).
- Therefore,
\[ 100^{17} \equiv 1^{17} \equiv 1 \mod 9. \]
- Now, calculate \( 100^{17} - 1 \):
\[ 100^{17} - 1 \equiv 1 - 1 \equiv 0 \mod 9. \]

2. **Calculate \( 10^{34} \mod 9 \)**:
- Next, simplify \( 10 \mod 9 \):
\[ 10 \div 9 = 1 \quad \text{(remainder 1)} \]
Thus, \( 10 \equiv 1 \mod 9 \).
- Therefore,
\[ 10^{34} \equiv 1^{34} \equiv 1 \mod 9. \]

3. **Combine the results**:
- Now substitute back into the expression:
\[ (100^{17} - 1) + (10^{34} + x) \equiv 0 + (1 + x) \mod 9. \]
- This simplifies to:
\[ 1 + x \mod 9. \]

4. **Set up the equation for divisibility by 9**:
- For the entire expression to be divisible by 9, we need:
\[ 1 + x \equiv 0 \mod 9. \]
- This means:
\[ x \equiv -1 \equiv 8 \mod 9. \]

5. **Conclusion**:
- Thus, the value of \( x \) that satisfies the condition is:
\[ x = 8. \]

### Final Answer:
The value of \( x \) is \( 8 \).

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