jmc

We want to find the smallest four-digit number that is a multiple of lcm[2, 3, 5, 7] = 210, so we need to find the smallest value of $n$ such that $$210n\ge 1000.$$Dividing this inequality by 210 we get $n\ge 4\frac{16}{21}$, so $n=5$ gives us the smallest four-digit multiple of 210: $210\cdot 5=\boxed{1050}$.