The least positive integer divisible by all the positive integers less than or equal to 10 is the $\(\operatorname{LCM}\)$ of $\(\{1,2,3,4,5,6,7$, $8,9\)$, and \(10$\}\)\(=2520=2^3 \times 3^2 \times 5 \times 7=K\)$.

Therefore, the total number of factors of $K$ (including 1 and $K)$ is $\((3+1)(2+1)(1+1)(1+1)=4 \times 3 \times 2 \times 2=48\)$.

Therefore, if we exclude 1 and $K$, the total number of factors turns out to be $\(48-2=46\)$.

Thus, the correct answer choice is $D$.