  Recently (in 2011) several new theorems concerning this conjecture were
proved by Bourgain and Kontorovich. The easiest of them states that the set of
numbers satisfying Zaremba's conjecture with A=50 has positive proportion in
$\N.$ The proof of this theorem is rather complicated and refers to the
spectral theory. In this paper,using only elementary methods, the same theorem
is proved with A=13 .
