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We obtain an upper bound for the number of solutions to the system of $m$ congruences of the type
modulo a prime $p$, with variables $1\leq x_{i}\leq h$, $i=1,\ldots ,{\it\nu}$ and arbitrary integers $s_{j},{\it\lambda}_{j}$, $j=1,\ldots ,m$, for a parameter $h$ significantly smaller than $p$. We also mention some applications of this bound.
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