For how many one-digit positive integers $k$ is the product $k \cdot 234$ divisible by 12 ?
Note that $234=9 \cdot 26=2 \cdot 3^{2} \cdot 13$.

Thus, $k \cdot 234=k \cdot 2 \cdot 3^{2} \cdot 13$.

This product is divisible by $12=2^{2} \cdot 3$ if and only if $k$ contributes another factor of 2 (that is, if and only if $k$ is even).

Since $k$ is a one-digit positive integer, then $k=2,4,6,8$.

Therefore, there are 4 one-digit positive integers $k$ for which $k \cdot 234$ is divisible by 12 .

ANSWER: 4