We obtain a new estimate for Kloosterman sum with primes $p\leqslant X$ to composite modulo $q$, that is, for the exponential sum of the type \[ \sum\limits_{ p\leqslant X,\;p\,\nmid q}\exp{ \biggl(\frac{ 2\pi i}{ q}\bigl(a\overline{ p}+bp\bigr)\,\biggr)},\quad (ab,q)=1,\quad p\overline{ p}\equiv 1\pmod{ q}, \] which is non-trivial in the case when $q^{ \,3/4+\varepsilon}\leqslant X\ll q^{ \,3/2}$. We also apply this estimate to the proof of solvability of some congruences with inverse prime residues $\pmod{ q}$.
Categories: math.NT