The prime factorization of 8 is $2^3$, the prime factorization of 40 is $2^3\cdot 5$, and the prime factorization of 16 is $2^4$. Taking the highest power of each prime that appears in the factorization of $8$, $40$, and $16$, we find that the greatest common factor is $2^3=\boxed{8}$.
