If a finite system of $r$ congruences $\{a_i(\mathrm{mod}{n}_i):1\le i\le r\}$ (the $n_i$ are not necessarily distinct) covers $2^r$ consecutive integers then it covers all integers.

AMS 11 · solved (literature)

theorem erdos_275 (r : ℕ) (a : Fin r → ℤ) (n : Fin r → ℕ)
    (H : ∃ k : ℤ, ∀ x ∈ Ico k (k + 2 ^ r), ∃ i, x ≡ a i [ZMOD n i]) (x : ℤ) :
    ∃ i, x ≡ a i [ZMOD n i]