  We give a simple proof of the well-known divisibility by 2 condition for
rational points on elliptic curves with rational 2-torsion. As an application
of the explicit division by $2^n$ formulas obtained in Sec.2, we construct
versal families of elliptic curves containing points of orders 4, 5, 6, and 8
from which we obtain an explicit description of elliptic curves over certain
finite fields $\mathbb{F}_q$ with a prescribed (small) group $E(\mathbb{F}_q)$.
In the last two sections we study 3- and 5-torsion.
  This paper supercedes arXiv:1605.09279 [math.NT] .
