  This is an English translation of the author's 1989 note in Russian,
published in a collection "Arithmetic and Geometry of Varieties" (V.E.
Voskresenski, ed.), Kuibyshev State University, Kuibyshev, 1989, pp. 57--67.
  Let $X$ be be an absolutely irreducible smooth projective surface over a
finite field $k$ of odd characteristic, let $Br(X)$ be the (commutative
periodic) Brauer group of $X$ and $DIV Br(X)$ the subgroup of its divisible
elements. We write $Br(X)_{DIV}$ for the quotient $Br(X)/DIV Br(X)$ and
$Br(X)_{DIV}(2)$ for its (finite) $2$-primary component. We prove that the
order of $Br(X)_{DIV}(2)$ is a full square under the following additional
assumptions on $\bar{X}=X\times \bar{k}$ where $ \bar{k}$ is an algebraic
closure of $k$.
  There is no 2-torsion in the N\'eron-Severi group of $\bar{X}$. The surface
$\bar{X}$ admits a lifting to characteristic 0.
  The proof is based on constructions of author's paper (Math. USSR Izv. 20
(1983), 203-234) and Wu's Theorem that relates Stiefel-Whitney classes and
Steenrod squares.
