If `(x)/(y)` has a remainder of 0 and `(z)/(y)` has a remainder of 3, what is the remainder of `(xz)/(y)`?

Answer

To solve the problem, we need to analyze the given conditions and apply the properties of divisibility.

### Step-by-Step Solution:

1. **Understanding the Given Conditions**:
- We know that \( \frac{x}{y} \) has a remainder of 0. This means that \( x \) is divisible by \( y \). We can express this mathematically as:
\[ x = k \cdot y \]
for some integer \( k \).

2. **Analyzing the Second Condition**:
- The second condition states that \( \frac{z}{y} \) has a remainder of 3. This means that when \( z \) is divided by \( y \), it can be expressed as:
\[ z = m \cdot y + 3 \]
for some integer \( m \). Here, \( z \) is not divisible by \( y \) since the remainder is 3.

3. **Finding \( xz \)**:
- Now, we want to find the remainder when \( xz \) is divided by \( y \). We can substitute the expressions for \( x \) and \( z \):
\[ xz = (k \cdot y)(m \cdot y + 3) \]
- Expanding this gives:
\[ xz = k \cdot y \cdot m \cdot y + k \cdot y \cdot 3 \]
- This simplifies to:
\[ xz = k \cdot m \cdot y^2 + 3k \cdot y \]

4. **Dividing \( xz \) by \( y \)**:
- Now, we divide \( xz \) by \( y \):
\[ \frac{xz}{y} = \frac{k \cdot m \cdot y^2 + 3k \cdot y}{y} \]
- This simplifies to:
\[ \frac{xz}{y} = k \cdot m \cdot y + 3k \]
- The term \( k \cdot m \cdot y \) is an integer (since it is multiplied by \( y \)), and thus contributes no remainder.

5. **Finding the Remainder**:
- The only term left that could contribute to the remainder is \( 3k \). However, since \( k \) is an integer, \( 3k \) is also an integer and when divided by \( y \), it does not affect the remainder from the \( z \) part.
- Since \( x \) is divisible by \( y \), the entire product \( xz \) will also have a remainder of 0 when divided by \( y \).

### Conclusion:
Thus, the remainder of \( \frac{xz}{y} \) is:
\[ \text{Remainder} = 0 \]