H5511 ** If a number can be decomposed into the product of $k$ consecutive natural numbers greater than 1, then we say this number has property $p(k)$.
(1) Find a $k$, for which there exists a number that has both properties $p(k)$ and $p(k+2)$;
(2) Prove: A number that has both properties $p(2)$ and $p(4)$ does not exist.
(1) Since $720=8 \cdot 9 \cdot 10=2 \cdot 3 \cdot 4 \cdot 5 \cdot 6$, it follows that $k=3$ satisfies the requirement.
(2) If $n(n+1)=m(m+1)(m+2)(m+3)$, then $n^{2}+n+1=\left(m^{2}+3 m+1\right)^{2}$, which is impossible. Because $n^{2}+n+1$ lies between two consecutive square numbers $n^{2}$ and $(n+1)^{2}$.