x, y and z are the natural numbers. Which of the following statements is true?
I. If x is divisible by y and y is divisible by z then x must be divisible by z.
II. If x is a factor of y and z, then x must be a factor of y + z.
III. If x is a factor of y and z, then x must be a factor of `(y)/(z)`.

Options

A. I, II and III
B. I only
C. I and II
D. II only

Answer

To determine which of the statements is true, we will analyze each statement one by one.

### Statement I:
**If x is divisible by y and y is divisible by z, then x must be divisible by z.**

1. **Understanding the statement**:
- If \( x \) is divisible by \( y \), it means \( x = k \cdot y \) for some integer \( k \).
- If \( y \) is divisible by \( z \), it means \( y = m \cdot z \) for some integer \( m \).

2. **Combining the two**:
- Substitute \( y \) in the first equation:
\[ x = k \cdot (m \cdot z) = (k \cdot m) \cdot z \]
- This shows that \( x \) is also divisible by \( z \) since \( k \cdot m \) is an integer.

3. **Conclusion**:
- Statement I is **true**.

### Statement II:
**If x is a factor of y and z, then x must be a factor of y + z.**

1. **Understanding the statement**:
- If \( x \) is a factor of \( y \), it means \( y = a \cdot x \) for some integer \( a \).
- If \( x \) is a factor of \( z \), it means \( z = b \cdot x \) for some integer \( b \).

2. **Combining the two**:
- Now, consider \( y + z \):
\[ y + z = (a \cdot x) + (b \cdot x) = (a + b) \cdot x \]
- Since \( a + b \) is an integer, \( y + z \) is also divisible by \( x \).

3. **Conclusion**:
- Statement II is **true**.

### Statement III:
**If x is a factor of y and z, then x must be a factor of (y/z).**

1. **Understanding the statement**:
- If \( x \) is a factor of \( y \), then \( y = a \cdot x \) for some integer \( a \).
- If \( x \) is a factor of \( z \), then \( z = b \cdot x \) for some integer \( b \).

2. **Analyzing \( y/z \)**:
- Now, consider \( y/z \):
\[ \frac{y}{z} = \frac{a \cdot x}{b \cdot x} = \frac{a}{b} \]
- This is not guaranteed to be an integer unless \( b \) divides \( a \). Therefore, \( x \) is not necessarily a factor of \( y/z \).

3. **Conclusion**:
- Statement III is **false**.

### Final Conclusion:
- The true statements are **I and II**.

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