Why do remainders show cyclic pattern? Let us find the remainders of $\dfrac{6^n}{7}$,
Remainder of $6^0/7 = 1$
Remainder of $6/7 = 6$
Remainder of $36/7 = 1$
Remainder of $216/7 = 6$
Remainder of $1296/7 = 1$
This pattern of $1,6,1,6...$ keeps on repeating. Why is it so? I'm asking in general, that is for every case of type $a^n/b$'s remainder keeps on repeating as we increase $n$.
P.S: This is a follow up question of my previous question.

Looking at remainders after division by 7 is called arithmetic modulo 7.
You are regarding powers of 6, modulo 7. But $6$ is $-1$, modulo $7$. This is written:
$$6 \equiv -1 \pmod 7$$
But the powers of $-1$, that is the numbers $(-1)^n$, simply alternate: $1,-1,1,-1,1,\ldots$.
