Question

What is the smallest positive integer $K$ such that $2000\times 2001\times K$ is a perfect cube?

• A. $2^3\times 3^3\times {23}^3\times {29}^3$
• B. $2\times 3\times 23\times 29$
• C. $2\times 3^2\times {23}^3\times {29}^4$
• D. $2^2\times 3^2\times {23}^2\times {29}^2$

Answer 1

Answer: (D)

$2^2\times 3^2\times {23}^2\times {29}^2$

Factors of $2000=2\times 2\times 2\times 2\times 5\times 5\times 5$
Factors of $2001=3\times 23\times 29$
For $2000\times 2001\times K$ to be a perfect cube, each factor must form triplets.
Thus, $K$ must be $2^2\times 3^2\times {23}^2\times {29}^2$