If x+9y is divisible by 5, where x and y are integers ,then what is the remainder when `8x+7y` is divided by 5 ?

Options

A. A. 2
B. B. 3
C. C. 1
D. D. 0

Answer

To solve the problem, we need to determine the remainder when \(8x + 7y\) is divided by 5, given that \(x + 9y\) is divisible by 5.

### Step-by-step Solution:

1. **Understanding the condition**: We know that \(x + 9y\) is divisible by 5. This means that:
\[ x + 9y \equiv 0 \mod 5 \]

2. **Simplifying the equation**: We can simplify \(9y\) modulo 5:
\[ 9 \equiv 4 \mod 5 \]
Thus, we can rewrite the condition as:
\[ x + 4y \equiv 0 \mod 5 \]
This implies:
\[ x \equiv -4y \mod 5 \]
or equivalently:
\[ x \equiv 1y \mod 5 \quad \text{(since } -4 \equiv 1 \mod 5\text{)} \]

3. **Substituting into the expression**: Now we need to find \(8x + 7y\) modulo 5. We can substitute \(x\) from the previous step:
\[ 8x + 7y \equiv 8(1y) + 7y \mod 5 \]
Simplifying this gives:
\[ 8y + 7y \equiv 15y \mod 5 \]

4. **Finding the remainder**: Now, we simplify \(15y\) modulo 5:
\[ 15 \equiv 0 \mod 5 \]
Therefore:
\[ 15y \equiv 0 \mod 5 \]

5. **Conclusion**: Since \(15y\) is divisible by 5, the remainder when \(8x + 7y\) is divided by 5 is:
\[ \text{Remainder} = 0 \]

### Final Answer:
The remainder when \(8x + 7y\) is divided by 5 is **0**.