Question

The smallest positive integer
K such the
cube
(2000) (2001) K 15. a
perfect​

Answer 1

Here

2000*2001*k is a perfect cube.

or 2^4*3*5^3*23*29*k us a perfect cube.

or (2^3*5^3) *2*3*23*29*k is a perfect cube.

After pairing up the factors we come to the conclusion that a2, a3, a23 and a 29 remain without being able to be paired into triplets.

Hence, if we complete the such triplets of the remaining unfortunately primes, we will find that

(2^3*5^3) *(2*2^2) *(3*3^2) *(23*23^2) *(29*29^2) becomes a perfect cube.

K=2^2*3^2*23^2*29^2=16016004

K*2000*2001=64096048008000