Solve the equation
(a^2,b^2)+(a,bc)+(b,ac)+(c,ab)=199. in positive integers.
(Here (x,y) denotes the greatest common divisor of x and y.)

(In reply to solution by xdog)

your error is in assuming (m,np)=(m,n)*(m,p)
A quick counter example is
m=2, n=4, p=8
(m,n)=(m,p)=2
(m,n)*(m,p)=4
however, (m,np)=2

I had a similar though as you of somehow simplifying the equation to be strictly in terms of (a,b), (a,c), and (b,c).  However, this boils down to finding (m,np) in terms of (m,n), (m,p), and (n,p).  Unfortunately I have not been able to find an equation that works in all cases. 

Posted by Daniel on 2013-03-15 02:02:43