Modular Equations - filtering the solutions

Question

I have a modular equation of the form:

$$3^a N + c ≡ 2^d \pmod {2^{d+1}}$$

where $a$, $c$ and $d$ are positive integers and $c ≡ 1 \pmod 6$ or $c ≡ 5 \pmod 6$

I know that these equations produce numbers that cycle 1, 3, and 5 (in some combination) when taken modulo 6. What I am interested in doing is modifying the equation somehow such that the only solutions are the multiples of 3.

A simple example:

$$ 3N + 1 ≡ 4 \pmod8$$

produces the sequence of solutions: $1, 9, 17, 25, 33, 41, 49, 57, 65, \ldots$

But I am only interested in: $9, 33, 57, 81, 105, \ldots$

By inspection I can see that this equation would work:

$$N ≡ 9 \pmod{24}$$

But I cannot see much logic behind it other than the $\bmod 24$ has come from $\pmod{3\times8}$

Any help greatly appreciated. Thanks in advance.