The four-digit numeral $3AA1$ is divisible by 9. What digit does $A$ represent?
To check if a number is divisible by 9, I can add up its digits and see if the sum is a multiple of 9. So, I need to find $A$ such that $3 + A + A + 1$ is a multiple of 9. That means $4 + 2A$ is a multiple of 9. The smallest multiple of 9 that is greater than 4 is 9 itself, so I can try $4 + 2A = 9$. Solving for $A$, I get $A = \frac{9 - 4}{2} = \frac{5}{2}$. But that's not a digit, so I need to look for the next multiple of 9. The next one is 18, so I can try $4 + 2A = 18$. Solving for $A$, I get $A = \frac{18 - 4}{2} = 7$. That is a digit, and it works for the problem. I can check by plugging it back into the original number and adding up the digits: $3 + 7 + 7 + 1 = 18$, which is a multiple of 9. So, the digit $A$ represents is 7. 