For each $m \in \{2, 3, 5\},$ the condition $a_{n+m} \equiv a_n \pmod{m}$ means the residue of $a_n$ modulo $m$ depends only on $n \bmod m,$ defining a map $\sigma_m$ from residues to residues. Each residue class of positions has $\frac{30}{m}$ members, and so does each residue class of values; if $\sigma_m$ sent two position classes to the same value class, that class's $\frac{30}{m}$ values would have to fill $\frac{60}{m}$ positions, which is impossible. So each $\sigma_m$ is a permutation of the residues modulo $m.$

Conversely, by the Chinese remainder theorem each position $n \in \{1, \ldots, 30\}$ corresponds to a unique triple $(n \bmod 2,\, n \bmod 3,\, n \bmod 5),$ and likewise for values. Any choice of permutations $(\sigma_2, \sigma_3, \sigma_5)$ therefore determines a unique valid permutation of $1, \ldots, 30,$ sending the position triple to the prescribed value triple.

Hence $N = 2! \cdot 3! \cdot 5! = 1440,$ and the remainder upon division by $1000$ is $440.$