306. We will say that a number has the property $\mathrm{P}(k)$ if it can be factored into the product of $k$ consecutive natural numbers, all greater than 1.

a) Find $k$ such that some number $N$ simultaneously has the properties $\mathrm{P}(k)$ and $\mathrm{P}(k+2)$.
b) Prove that there are no numbers that simultaneously have the properties $P(2)$ and $\mathbf{P}(4)$.
306. a) $k=3 ; 720=2 \cdot 3 \cdot 4 \cdot 5 \cdot 6=8 \cdot 9 \cdot 10$.

b) If $m(m+1)=n(n+1)(n+2)(n+3)$, then $m^{2}+m+1=\left(n^{2}+3 n+1\right)^{2}$, which is impossible.