Problem 335

Suppose that the length, width and height of the prism are the positive integers $a$, $b$ and $c$.

Since the volume of the prism is 21, then $abc=21$.

We note that each of $a$, $b$ and $c$ is a positive divisor of 21.

The positive divisors of 21 are 1, 3, 7, and 21, and the only way to write 21 as a product of three different integers is $1 \times 3 \times 7 = 21$.

Therefore, the length, width and height of the prism must be 1, 3, and 7, in some order.

The sum of these is $1+3+7=11$.