I often deal with combinatorial numbers $x$ are difficult to compute exactly, but it's possible to find congruences satisfied by them. You can combine the congruences using the Chinese Remainder Theorem into one concise congruence. If one day someone does compute the number $x$ by a lengthy computation, they can check that $x$ satisfies the congruence.

For instance, the number of reduced Latin squares $R_{12}$ of order $12$ is unknown, but it satisfies $R_{12} \equiv 50400 \pmod {55440}$.