What is the algebraic structure describing divisibility of residue class products with powers of two, e.g. $7\mid[1]_7\cdot2^{3n+3}-1$? Let $n$ be an integer $n=0,1,2,\ldots$ The divisibility of the following odd numbers (e.g. by 7) is structured as follows:

*

*$7\mid[4]_7\cdot2^{3n+1}-1$

*$7\mid[2]_7\cdot2^{3n+2}-1$

*$7\mid[1]_7\cdot2^{3n+3}-1$
What is the general law for such divisibilities? Which algebraic structure (ideals, rings, p-adic valuations, or whatever) covers such behavior?
May even I missed some residue classes - how may I show that the three above-shown cases cover all such divisibilities?
The same applies to the divisibility by five:

*

*$5\mid[3]_5\cdot2^{4n+1}-1$

*$5\mid[4]_5\cdot2^{4n+2}-1$

*$5\mid[2]_5\cdot2^{4n+3}-1$

*$5\mid[1]_5\cdot2^{4n+4}-1$
Of course I can show inductively that these divisibilities exists. But is there a general algebraic approach that explains this?
 A: The set of residues mod $n$ is a ring, with the usual (modular) addition and multiplication. For example, your first divisibility relation can be written as
$$2^{3n+1}\cdot 4 \equiv1\pmod{7},$$
which is easily verified for all $n$ because $2^{3n}\equiv8^n\equiv1^n\equiv1\pmod{7}$.
A: Your divisibilities are immediate consequences of integer exponent laws and CPR =  Congruence Power Rule, when viewed arithmetically in terms of congruences, namely they have  the form
$\!\!\bmod 5\!:\,\ 2^{\large J} 2^{\large 4N+4-J}= 2^{\large 4N+4}= (\color{#c00}2^{\large 4})^{\large N+1}\equiv \color{#c00}1^{\large N+1}\equiv 1\ $ by $\,\color{#c00}{2^{\large 4}\equiv 1}\,$ and CPR
e.g. for $\,J=3\,$ we get $\,1\equiv \color{#0a0}{2^{\large 3}} 2^{\large 4N+1}\equiv \color{#0a0}{3}\cdot 2^{\large4 N+1}$ as in your list, by $\,\color{#0a0}{2^{\large 3}\equiv 3}$.
Essentially you have discovered the cyclic group structure of $\langle 2\rangle $ = $\{ 2^n\ :\ n\in \Bbb N\}.\,$ The same cyclicity will be true of $\langle a\rangle$ for any $a$ coprime to the modulus $m$ using $\,a^{\phi(m)} \equiv 1\pmod{\!m}\,$ by Euler (or $\,a^n\equiv 1\,$ for $\,n\,$ being the order of $\,a,\,$ which divides $\,\phi(m)\,$ by the Order Theorem).
Such divisibility properties are further algebraically reified by passing from congruence arithmetic to the corresponding quotient ring $\Bbb Z/5$. Then the above congruences become equalities when the integers are interpreted as names for their equivalence class, e.g. $\,2\,$ denotes $\,[2]_5 = 2 + 5 \Bbb Z$.
