Proof for factors of a number I was trying to prove the following:

  if  x%(x/2) != 0 or x%(x/2) == 0 

  then x%(x/y) != 0 or x%(x/y) == 0  such that y = [2,4)


So I am trying to figure out whether a number can have a factor in between its half and its fourth.  I am trying to use it in a factoring algorithm.
If a number $n$ has a factor $m$ in the range $n/4 \leq m \leq n/2$, then the quotient $k = n/m$ satisfies $2 \leq k \leq 4$, that is, $k \in \{2,3,4\}$. This should give you a fast algorithm for determining whether such a factor $m$ exists.