$Between three consecutive numbers at least one must be even and one must be a multiple of$\;3.
n(n+1)(n+2)
$Say$\; n \;$was the even one and$\; (n+1)\; $was the one divisible by$\; 3. \;$Then we can write$\; n = 2j\; $for integer$\; j \;$and$\; n+1 = 3m\; $for some integer$\; m. \;$In all...