jmc

Writing out the prime factors of $22$ and $48$, we see that $22=2\cdot 11$ and $48=16\cdot 3=2^4\cdot 3$. Since $11$ does not divide $48$ and $3$ does not divide $22$, it follows that the greatest common factor of $22$ and $48$ is $2$.

The least common multiple, on the other hand, must account for the prime factors of both $22$ and $48$. The highest power of $2$ present in either term is $2^4=16$, of $3$ is $3^1=3$, and of $11$ is $11^1=11$. Thus, the LCM will be $2^4\cdot 3\cdot 11=528$. The product of the greatest common factor and the least common multiple is thus $2\cdot 528=\boxed{1056}$.