On a modular form of Zaremba's conjecture

We prove that for any prime $p$ there is a divisible by $p$ number $q = O(p^{30})$suchthatforacertainpositiveinteger$a$coprimewith$q$ the ratio $a\mathrm{/}q$ has bounded partial quotients. In the other direction we show that there is an absolute constant $C>0$ such that for any prime $p$ exist divisible by $p$ number $q=O(p^C)$ and a number $a$, $a$ coprime with $q$ such that all partial quotients of the ratio $a\mathrm{/}q$ are bounded by two.