Creative Problem Solving

[T0rajir0u]

4
6
8
(10 don't work cuz 9 is a remainder)
12
18
24
30

(yeah we want multiples of 6, but 36 doesn't work because 25 is a viable remainder)

there's probably some bounding argument i could use to prove that no others exist but i'm way too lazy

edit: CLAIM: no numbers larger than 30 work

we need multiples of 30. otherwise, either 4, 9, or 25 are viable remainders

however if we try 60 then 49 is a viable remainer so we need multiples of 210

but then 121 is a viable remainder etc

the sequence of products of primes here that we need grows faster than the sequence of squares of primes (WAY too lazy to prove this) so nothing above 30 works