  The number-theoretic codes are a class of codes defined by single or multiple
congruences and are mainly used for correcting insertion and deletion errors.
Since the number-theoretic codes are generally non-linear, the analysis method
for such codes is not established enough. The distance enumerator of a code is
a unary polynomial whose $i$th coefficient gives the number of the pairs of
codewords with distance $i$. The distance enumerator gives the maximum
likelihood decoding error probability of the code. This paper presents an
identity of the distance enumerators for the number-theoretic codes. Moreover,
as an example, we derive the Hamming distance enumerator for the
Varshamov-Tenengolts (VT) codes.
