B2. The product $8000 \times K$ is a square, where $K$ is a positive integer.
What is the smallest possible value of $K$ ?
SolUtion
Note first that $8000=8 \times 1000=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5$.
For $K$ as small as possible, $8000 \times K$ must be as small as possible.
Since $8000 \times K$ is a square, write $8000 \times K=\left(2^{3} \times 5 \times 5\right) \times\left(2^{3} \times 5 \times K\right)$.
Therefore, the smallest possible value of $K$ is 5 .