For these statements, the following is true : x is divisible by 24. For each statement, say whether it must be true, could be true, or cannot be true.
x is divisble by 9.

Answer

To determine whether the statement "x is divisible by 9" is true, given that "x is divisible by 24," we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Divisibility**:
- We know that if x is divisible by 24, it means that x can be expressed as \( x = 24n \), where n is an integer. This implies that x leaves no remainder when divided by 24.

2. **Express x in terms of n**:
- From the above, we can write \( x = 24n \).

3. **Check divisibility by 9**:
- We need to check if \( x \) is divisible by 9. This means we need to see if \( \frac{x}{9} \) is an integer.
- Substitute \( x = 24n \) into this expression:
\[ \frac{x}{9} = \frac{24n}{9} \]

4. **Simplifying the expression**:
- Simplifying \( \frac{24n}{9} \) gives:
\[ \frac{24n}{9} = \frac{8n}{3} \]
- For \( \frac{8n}{3} \) to be an integer, \( n \) must be a multiple of 3 because 3 is in the denominator.

5. **Testing values of n**:
- If \( n = 0 \):
\[ \frac{8 \cdot 0}{3} = 0 \quad \text{(divisible by 9)} \]
- If \( n = 1 \):
\[ \frac{8 \cdot 1}{3} = \frac{8}{3} \quad \text{(not divisible by 9)} \]
- If \( n = 2 \):
\[ \frac{8 \cdot 2}{3} = \frac{16}{3} \quad \text{(not divisible by 9)} \]
- If \( n = 3 \):
\[ \frac{8 \cdot 3}{3} = 8 \quad \text{(not divisible by 9)} \]
- If \( n = 6 \):
\[ \frac{8 \cdot 6}{3} = 16 \quad \text{(not divisible by 9)} \]

6. **Conclusion**:
- Since \( \frac{8n}{3} \) is an integer only when \( n \) is a multiple of 3, we conclude that x could be divisible by 9 for certain values of n (like n=0, n=3, etc.) but not for others (like n=1, n=2).
- Therefore, the statement "x is divisible by 9" **could be true**.