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.

to make it a square try looking at product of means and product of extremes

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

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

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

X*(X+2) but you need this product to equal perfect square

so

x^2+2x=z^2

X^2+2x-z^2=0

look at discriminant b^2-4ac=4-4z^2which is only positive if Z=0,1 which you can prove can't be achieved for x(x+2)=z!!!!

 

 

 

 

 

Posted by red_sox_fan_032003 on 2004-02-25 17:42:49