AUSTRIAN-POLISH 2006 Problem 1

Let $M(n)=\{n,n+1,n+2,n+3,n+4,n+5\}$ be a set of 6 consecutive integers. Let's take all values of the form $$\frac{a}{b}+\frac{c}{d}+\frac{e}{f}$$ with the set $\{a,b,c,d,e,f\}=M(n)$ . Let $$\frac{x}{u}+\frac{y}{v}+\frac{z}{w}=\frac{xvw+yuw+zuv}{uvw}$$ be the greatest of all these values.

a) show: for all odd $n$ hold: $\gcd (xvw+yuw+zuv, uvw)=1$ iff $\gcd (x,u)=\gcd (y,v)=\gcd (z,w)=1$ . b) for which positive integers $n$ hold $\gcd (xvw+yuw+zuv, uvw)=1$ ?