On multiplicative congruences

Let ${varepsilon}$ 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 ${N_1N_2N_3 > m^{1+varepsilon}, }$ the set $${x_1x_2x_3 quad ({rm mod},m): quad x_jin [1,N_j]}$$ 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 ${ N_1N_2N_3N_4 > m^{1+varepsilon}, }$ the set $${x_1x_2x_3x_4 quad ({rm mod},m): quad x_jin [1,N_j]}$$ also contains almost all the residue classes modulo m. Let p be a large prime parameter and let ${p > N > p^{63/76+varepsilon}.}$ We prove that for any nonzero integer constant k and any integer ${lambdanotequiv 0 ,, ({rm mod},p)}$ the congruence $$p_1p_2(p_3+k)equiv lambda quad ({rm mod}, p) $$ admits (1 + o(1))π(N)³/p solutions in prime numbers p ₁, p ₂, p ₃ ≤ N.