$\definecolor{A}{RGB}{121,96,253}\color{A}\fbox{Solution 1.}$ Observe that $$7|N+2\wedge8|N+2\wedge9|N+2.$$Smallest $N+2$ satisfying this condition is $$N+2=7\cdot8\cdot9.\blacksquare$$$\definecolor{A}{RGB}{121,96,253}\color{A}\fbox{Solution 2.}$ $$7| N-5,\ 8| N-6\implies 56| 8(N-5)-7(N-6)=N+2\implies \exists_{k\in\mathbb{Z}}\left(k\ge0\wedge N=56k +54\right)$$Easy to see $$\exists_{k\in\mathbb{Z}}\left(k\ge0\wedge N=56k +54\right)\implies\left(7| N-5\wedge 8| N-6\right)$$$$9|(56k +54)-7\iff 9|2k+2\iff 9|k+1$$Smallest $k$ is $8$. Therefore $$N=56\cdot 8+54=502.$$#1886