What is the smallest integer $b > 3$ for which the base $b$ number $23_b$ is a perfect square?
I can rewrite $23_b$ as $2b + 3$ in base 10, since the digits in base $b$ represent powers of $b$. I can try different values of $b$ and see if they work. If $b = 4$, then $2b + 3 = 2 \cdot 4 + 3 = 11$, which is not a perfect square. If $b = 5$, then $2b + 3 = 2 \cdot 5 + 3 = 13$, which is not a perfect square. If $b = 6$, then $2b + 3 = 2 \cdot 6 + 3 = 15$, which is not a perfect square. If $b = 11$, then $2b + 3 = 2 \cdot 11 + 3 = 25$, which is a perfect square, since $25 = 5^2$. Therefore, the smallest base $b > 3$ that makes $23_b$ a perfect square is $b = 11$. 