  Let $\epsilon$ be a fixed positive quantity, $m$ be a large integer, $x_j$
denote integer variables. We prove that for any positive integers $N_1,N_2,N_3$
with $N_1N_2N_3>m^{1+\epsilon},$ the set $$ \{x_1x_2x_3 \pmod m: \quad x_j\in
[1,N_j] \} $$ contains almost all the residue classes modulo $m$ (i.e., its
cardinality is equal to $m+o(m)$). We further show that if $m$ is cubefree,
then for any positive integers $N_1,N_2,N_3,N_4$ with
$N_1N_2N_3N_4>m^{1+\epsilon},$ the set $$ \{x_1x_2x_3x_4 \pmod m: \quad x_j\in
[1,N_j] \} $$ also contains almost all the residue classes modulo $m.$
  Let $p$ be a large prime parameter and let $p>N>p^{63/76+\epsilon}.$ We prove
that for any nonzero integer constant $k$ and any integer $\lambda\not\equiv
0\pmod p$ the congruence $$ p_1p_2(p_3+k)\equiv \lambda\pmod p $$ admits
$(1+o(1))\pi(N)^3/p$ solutions in prime numbers $p_1, p_2, p_3\le N.$
