Given the following family of hash functions:
$$ \mathbb{H} = \{h_c(x) = (12x + c) \bmod m \mid c \in \mathbb{N} \}, $$ where $m$ is the key size.
Prove that $\mathbb{H}$ is not a universal family of hash functions.
So I got the following idea:
If $m = 12 $ then $H$ is not an universal family of hash functions. But is this enough or have I to prove it for every/random m?
This leads to the question: Is a universal family of hash functions dependent on the hash table size?