Actually, the conclusion you've drawn about the remainder when \( 4x \) is divided by 9 contains an error. Let's clarify the situation step by step.
Starting with the information given:
- When \( x \) is divided by 9, the remainder is 3. This means: \[ x = 9k + 3 \quad \text{for some integer } k. \]
Now, we multiply both sides by 4: \[ 4x = 4(9k + 3) = 36k + 12. \]
Next, we need to find the remainder when \( 4x \) is divided by 9. To do this, we can consider the expression \( 36k + 12 \).
Now, observe that:
- \( 36k \) is divisible by 9 (since \( 36 \div 9 = 4 \)).
- The term \( 12 \) gives a remainder when divided by 9.
Now, let's divide 12 by 9: \[ 12 \div 9 = 1 \quad \text{with a remainder of } 3. \]
Thus, when \( 4x = 36k + 12 \) is divided by 9, only the term \( 12 \) contributes to the remainder, and we find that: \[ \text{Remainder} = 3. \]
So, when \( 4x \) is divided by 9, the remainder is correctly calculated as 3. Your calculation was mostly correct, but the conclusion about me reiterating how to correctly think about the sums and divisions properly illustrates the step-by-step nature of finding remainders.
Final Summary: The remainder when \( 4x \) is divided by 9 is indeed 3.