Let n be the smallest positive integer such that n(n+1)(n+2)(n+3) can be expressed as either a perfect square or a perfect cube (not necessarily both).

Find n, or prove that this is not possible.

try looking at product of means and product of extremes

N(N+3),(N+1)(N+2)

n^2+3n,n^2+3n+2

now subbing in X=n^2+3n you have

X*(X+2) but you need this product to equal perfect square,perfect cube the only common factor they can have is 2.

If X is odd it is easy to show that they have no common factors therefore since X and X+2 can't both be perfect squares/cubes there is no solution.

So this leaves case of X is even