Every nonzero integer
has an inverse (modulo )
for
a prime and
not a multiple of .
For example, the modular inverses of 1, 2, 3, and 4 (mod 5) are 1, 3, 2, and 4.

If
is not prime, then not every nonzero integer has a modular inverse. In fact, a nonzero integer has a modular inverse modulo iff and are relatively prime.
For example,
(mod 4) and
(mod 4), but 2 does not have a modular inverse.

The triangle above (OEIS A102057) gives modular inverses of
(mod )
for ,
2, ...,
and ,
3, .... 0 indicates that no modular inverse exists.

If
and
are relatively prime, there exist integers and such that , and such integers may be found using the Euclidean
algorithm. Considering this equation modulo , it follows that ; i.e., .