Pairs of Integers whose Product with Divisor Count are Equal

Theorem
Let $\map {\sigma_0} n$ denote the divisor counting function: the number of divisors of $n$.

The following pairs of integers $T$ have the property that $m \, \map {\sigma_0} m$ is equal for each $m \in T$:
 * $\set {18, 27}$
 * $\set {24, 32}$
 * $\set {56, 64}$