# Remarkable subsets

Let $$A \subset \mathbb Z$$ be a non-trivial set of integers $$($$i.e. $$A \not= \emptyset$$ and $$A \not= \mathbb Z).$$ We call such a set $$A$$ a "remarkable set of type $$N$$" if it has the following properties:

• If $$a$$ is an element of $$A,$$ then $$-a$$ is also an element of $$A$$.

• If $$a$$ is an element of $$A$$, then $$a+N$$ is also an elements of $$A$$.

• If $$a,b$$ are elements of $$A$$ (not necessarily different), then $$a + 2b$$ is also an element of $$A$$.

How many (non-trivial) remarkable sets of type 18 are there?

Bonus: Generalize. If $$N$$ = $$2^{e_2}\cdot 3^{e_3}\cdot 5^{e_5}\cdots$$ is the prime factorization of $$N$$, how many non-trivial remarkable sets of type $$N$$ exist? What do they look like?

×