What is the product of the least common multiple and the greatest common factor of 22 and 48?
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 $2 \cdot 528 = \boxed{1056}$.