The expression "A - 300 is a multiple of 3 but not 9" implies certain properties about the value of \( A \). Here's a step-by-step explanation:

1. **Multiple of 3 but not 9 condition**:
   For \( A - 300 \) to be divisible by 3 but **not** divisible by 9, the number \( A - 300 \) must satisfy the divisibility rule for 3 but fail the divisibility rule for 9. 

   - A number is divisible by 3 if the sum of its digits is divisible by 3.
   - A number is divisible by 9 if the sum of its digits is divisible by 9.

2. **What this implies about A**:
   - Since \( A - 300 \) is divisible by 3, \( A \equiv 300 \ (\text{mod} \ 3) \).
   - This means \( A \equiv 0 \ (\text{mod} \ 3) \), so \( A \) must also be divisible by 3.
   - However, \( A - 300 \) is not divisible by 9, which implies that while \( A \) must still follow the divisibility by 3 rule, it does not meet the stricter condition of divisibility by 9.

Thus, \( A - 300 \) must be a multiple of 3, and \( A \) must be chosen such that when you subtract 300 from \( A \), the result satisfies the condition of being divisible by 3 but not by 9.

### To summarize:
- \( A - 300 \) is divisible by 3 but not divisible by 9.
- \( A \) is divisible by 3, but \( A - 300 \) is not divisible by 9.

### Example:
Let’s assume \( A = 303 \):
- \( A - 300 = 303 - 300 = 3 \), which is divisible by 3 but not by 9.
Thus, \( A = 303 \) satisfies the given condition.

Would you like more details on this, or do you have any other questions?

Here are 5 related questions to expand your understanding:
1. What is the divisibility rule for a number to be divisible by 3?
2. What is the divisibility rule for a number to be divisible by 9?
3. Can \( A - 300 \) ever be divisible by 9? Why or why not?
4. How would the result change if the condition was \( A - 300 \) being a multiple of both 3 and 9?
5. Can you find another value of \( A \) that satisfies this condition?

**Tip:** To determine divisibility by 3 and 9, always check the sum of the digits of the number!

Explore the properties of divisibility for A - 300, which is divisible by 3 but not by 9. Understand the key concepts of modular arithmetic and divisibility rules for 3 and 9. This example problem is suitable for students in Grades 8-10, demonstrating a practical application of number theory.

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