An old question of Erdős asks if there exists, for each number
N
N
, a finite set
S
S
of integers greater than
N
N
and residue classes
r
(
n
)
(
mod
n
)
r(n)~(\textrm {mod}~n)
for
n
∈
S
n\in S
whose union is
Z
\mathbb Z
. We prove that if
∑
n
∈
S
1
/
n
\sum _{n\in S}1/n
is bounded for such a covering of the integers, then the least member of
S
S
is also bounded, thus confirming a conjecture of Erdős and Selfridge. We also prove a conjecture of Erdős and Graham, that, for each fixed number
K
>
1
K>1
, the complement in
Z
\mathbb Z
of any union of residue classes
r
(
n
)
(
mod
n
)
r(n)~(\textrm {mod}~n)
, for distinct
n
∈
(
N
,
K
N
]
n\in (N,KN]
, has density at least
d
K
d_K
for
N
N
sufficiently large. Here
d
K
d_K
is a positive number depending only on
K
K
. Either of these new results implies another conjecture of Erdős and Graham, that if
S
S
is a finite set of moduli greater than
N
N
, with a choice for residue classes
r
(
n
)
(
mod
n
)
r(n)~(\textrm {mod}~n)
for
n
∈
S
n\in S
which covers
Z
\mathbb Z
, then the largest member of
S
S
cannot be
O
(
N
)
O(N)
. We further obtain stronger forms of these results and establish other information, including an improvement of a related theorem of Haight.