A unit of O is an element with a multiplicative inverse still in O. If x has nonzero residue modulo two, then v_2(x)=0. Since K is a field, x has an inverse in K. Multiplicativity gives 0=v_2(1)=v_2(x)+v_2(x^(-1)), so its inverse also has valuation zero. Every coordinate of that inverse therefore has nonnegative valuation, placing it in O. Conversely, if x is divisible by two, its inverse would have negative valuation and cannot be in O. Nonzero x alone is insufficient; its residue must be nonzero.