### Instruction: For how many values of the digit $A$ is it true that $63$ is divisible by $A$ and $273{,}1A2$ is divisible by $4$?### Response: To start, I notice that the divisibility by $4$ rule depends only on the last two digits of a number, so I can ignore the first four digits of $273{,}1A2$ and just focus on $A2$.
I also recall that a number is divisible by $4$ if and only if its last two digits form a multiple of $4$.
So, I need to find the values of $A$ that make $A2$ a multiple of $4$.
I can list the multiples of $4$ that end with $2$: $12, 32, 52, 72, 92$.
These correspond to the values of $A$: $1, 3, 5, 7, 9$.
However, I also need to check if these values of $A$ satisfy the other condition, that $63$ is divisible by $A$.
I know that $63$ is divisible by $3, 7, 9$, but not by $1$ or $5$.
Therefore, the only values of $A$ that work are $3, 7, 9$.
There are three such values.