What is the least positive integer divisible by each of the first eight positive integers?
To find the least common multiple of the positive integers less than or equal to 8, we prime factorize each of them. \begin{align*}
2 &= 2 \\
3 &= 3 \\
4 &= 2^2 \\
5 &= 5 \\
6 &= 2\cdot 3 \\
7 &= 7 \\
8 &= 2^3.
\end{align*} Taking the maximum exponent for each prime, we find a least common multiple of $2^3\cdot3\cdot5\cdot7=(2\cdot5)\cdot 2^2\cdot3\cdot7=10\cdot84=\boxed{840}$.