jmc

Simplify $(2+3{)}^{23}=5^{23}$. Since the ones digit of $5\times 5$ is 5, the ones digit of $5^n$ is 5 for any positive integer $n$. Similarly, since the tens digit of $25\times 5$ is 2 (and the ones digit is 5), the tens digit of $5^n$ is 2 for all positive integers $n\ge 2$. Therefore, the sum of the tens digit and the ones digit of $(2+3{)}^{23}$ is $2+5=\boxed{7}$.