jmc

To determine a remainder when a number is divided by $5$, you only need to look at the last digit. If the last digit is $0$ or $5$, the remainder is $0$. If the last digit is $1$ or $6$, the remainder is $1$, and so on. To determine the last digit of $1492\cdot 1776\cdot 1812\cdot 1996$, you only need to look at the last digit of each number in the product. Thus, we compute $2\cdot 6\cdot 2\cdot 6=12^2=144$. The last digit of the number $1492\cdot 1776\cdot 1812\cdot 1996$ is also $4$, and thus the remainder when the number is divided by $5$ is also $4$, which gives an answer of $\boxed{4}$.