Find all $a$ such that for any positive integer $n$, the number $an(n+2)(n+3)(n+4)$ is an integer. (Author: O. Podlipski)
(similar to Problem 5 of grade 9)
Same problem for grades 10 and 11
$ n=1 $ then $ 60a $ is an integer so $ a=t/60 $ with $ t $ integer.
for $ n=4 $ then $ 4*6*7*8 $ is not divisible by $ 5 $ so $ a=w/12 $.
if $ n=3 $ then $ 3*5*6*7 $ is not divisible by $ 4 $ so $ a=z/6 $ with $ z $ an integer.
the number then is an integer for each $ n $ because $ (n+2)(n+3)(n+4) $ is divisible by $ 6 $.