2. If $2 A 99561$ is equal to the product when $3 \times(523+A)$ is multiplied by itself, find the digit $A$.
2. 4
We are given that $2 A 99561=[3 \times(523+A)]^{2}$, which is equivalent to $2 A 99561=9 \times(523+A)^{2}$. Since $(523+A)^{2}$ is an integer, it follows that $2 A 99561$ is divisible by 9 . By the rule on divisibility by 9 , after adding all the digits of $2 A 99561$, it suffices to find the digit $A$ for which $A+5$ is divisible by 9 , which yields $A=4$.