  Let $b(n)$ denote the number of cubic partition pairs of $n$. We give
affirmative answer to a conjecture of Lin, namely, we prove that
$$b(49n+37)\equiv 0 \pmod{49}.$$ We also prove two congruences modulo $256$
satisfied by $\overline{b}(n)$, the number of overcubic partition pairs of $n$.
Let $\overline{a}(n)$ denote the number of overcubic partition of $n$. For a
fixed positive integer $k$, we further show that $\overline{b}(n)$ and
$\overline{a}(n)$ are divisible by $2^k$ for almost all $n$. We use arithmetic
properties of modular forms to prove our results.
