jmc

We have $840=2^3\cdot 3\cdot 5\cdot 7$. From this prime factorization, it is clear that the product of four consecutive positive integers is $840=2^2\cdot 5\cdot (2\cdot 3)\cdot 7=4\cdot 5\cdot 6\cdot 7$. The largest of the four integers is $\boxed{7}$.