To find the least common multiple $LCM$ of $3m$ and $3$, we first need to factorize each number.

For $3m$, the factors are $3$ and $m$.
For $3$, the only factor is $3$.

The LCM is the product of the highest power of each prime factor. Therefore, the LCM of $3m$ and $3$ is $3\times m=3m$.

So, the least common multiple of $3m$ and $3$ is $3m$.