Problem 1

We see that $1^2=1$, $2^2=4$, $5^2=25$, and $4^2=16$, so already we know that either $\text{E}$ is the answer or the problem has some issues.
For integers, only the units digit affects the units digit of the final result, so we only need to test the squares of the integers from $0$ through $9$ inclusive. Testing shows that $8$ is unachievable, so the answer is $\boxed{\text{E}}$.