What is the smallest positive integer $p$ for which $\sqrt{2^{3} \times 5 \times p}$ is an integer?
(A) 2
(B) 5
(C) 10
(D) 1
(E) 20
If $\sqrt{2^{3} \times 5 \times p}$ is an integer, then $2^{3} \times 5 \times p$ is a perfect square.

For $2^{3} \times 5 \times p$ to be a perfect square, each prime factor must occur an even number of times.

For $p$ to be as small as possible, $p$ must have at least one factor of 2 and at least one factor of 5 .

Thus, the smallest possible value of $p$ is 10 .

(Checking, $2^{3} \times 5 \times 10=400$, which is a perfect square.)

ANswer: (C)