GCD Proof

Nov 2009
80
6
I have a hunch that this problem is solved using using the theorem (a +cb, b) = (a, b) where a, b, and c are all integers (a and b are not both zero).

Show that if n is a positive integer, then \(\displaystyle (n^2 + 2, n^3 +1) = \) 1, 3, or 9.

I just can't get this to work out.
 
Apr 2012
54
12
Kanpur,India
\(\displaystyle (n^2+2,n^3+1)=(n(n^2+2),n^3+1)=(n^2+2,2n-1)\)
\(\displaystyle =(n^2+2,n(2n-1))=(2n-1,n+4)=(n+4,9)=1,3,9\)