When positive integer 'X' is divided by 9, the remainder is 6. What is the remainder when X8 is divided by 9?

Options

A. `-1`
B. 8
C. 6
D. 3

Answer

To solve the problem, we need to find the remainder when \(8X\) is divided by 9, given that when \(X\) is divided by 9, the remainder is 6.

### Step-by-Step Solution:

1. **Understand the given information**:
We know that when \(X\) is divided by 9, the remainder is 6. This can be expressed mathematically as:
\[ X \equiv 6 \mod 9 \]

2. **Multiply \(X\) by 8**:
We need to find \(8X\). If we multiply both sides of the congruence by 8, we get:
\[ 8X \equiv 8 \times 6 \mod 9 \]

3. **Calculate \(8 \times 6\)**:
Now, calculate \(8 \times 6\):
\[ 8 \times 6 = 48 \]

4. **Find the remainder when 48 is divided by 9**:
Next, we need to find the remainder of 48 when divided by 9. We can do this by performing the division:
\[ 48 \div 9 = 5 \quad \text{(which gives a quotient of 5)} \]
Now, calculate \(9 \times 5\):
\[ 9 \times 5 = 45 \]
Now, subtract this from 48 to find the remainder:
\[ 48 - 45 = 3 \]

5. **Conclusion**:
Therefore, the remainder when \(8X\) is divided by 9 is:
\[ 8X \equiv 3 \mod 9 \]

### Final Answer:
The remainder when \(8X\) is divided by 9 is **3**.

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