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!