Which is the largest integer whose square root is an irrational number between 3 and 4?

Answer 1

The largest integer whose square root is an irrational number between 3 and 4 is $\boxed{15}$. Since $3^2=9$ and $4^2=16$, the only remaining possibility is $\sqrt{16}=4$, which is not an irrational number. To show that $\sqrt{15}$ is between 3 and 4, we note that $3^2=9<15<16=4^2$.