jmc

To find the least common multiple of the positive integers less than or equal to 8, we prime factorize each of them. $$\begin{array}{rl}2& =2\\ 3& =3\\ 4& =2^2\\ 5& =5\\ 6& =2\cdot 3\\ 7& =7\\ 8& =2^3.\end{array}$$ Taking the maximum exponent for each prime, we find a least common multiple of $2^3\cdot 3\cdot 5\cdot 7=(2\cdot 5)\cdot 2^2\cdot 3\cdot 7=10\cdot 84=\boxed{840}$.