On multiplicative congruences

Let $ϵ$ 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 $N_1{N}_2{N}_3>m^{1+ϵ},$ the set $$\{x_1{x}_2{x}_3(\mathrm{m}\mathrm{o}\mathrm{d}m):x_j\in [1,N_j]\}$$ 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 $N_1{N}_2{N}_3{N}_4>m^{1+ϵ},$ the set $$\{x_1{x}_2{x}_3{x}_4(\mathrm{m}\mathrm{o}\mathrm{d}m):x_j\in [1,N_j]\}$$ also contains almost all the residue classes modulo $m\mathrm{.}$ Let $p$ be a large prime parameter and let $p>N>p^{63\mathrm{/}76+ϵ}\mathrm{.}$ We prove that for any nonzero integer constant $k$ and any integer $\lambda\not\equiv 0\pmod p$thecongruence$$p_1{p}_2(p_3+k)\equiv \lambda (\mathrm{m}\mathrm{o}\mathrm{d}p)$$admits$(1+o(1))\pi(N)^3/p$solutionsinprimenumbers$p_1, p_2, p_3\le N.$