# Minimization of linear combination of vectors

Suppose I have a vectors in $\mathbb{R}^{6}$. For example, take the vector $v = k(-2,2,4,6,2,2)$. I would like to find some $k$ with the constraint that the last two components are necessarily non-negative and that every component in the vector is a minimal integer. For example, in this case, I would take $k = \frac{1}{2}$. Is there a way to implement such a minimization procedure in mathematica given an arbitrary six-component vector?

In particular, suppose I have now $n$ such vectors with $n \lt 6$. Is there a way to find the linear combination of such vectors with minimal integer entries and with the same constraint that the last two components are non-negative?

I've thought about trying to use Reduce or Solve, but I was wondering if there's some faster way to implement this.

Thanks!

f[v_] := v Sign[v[[-1]]]/GCD @@ v