<s>[INST] Find the unit digit of $2124^{392}+3143^{394}*7177^{392}-8818^{394}$ What I do here is I find the remainder of each parcel in mod 10 and then do the maths and mod 10 again:
$2124^{392} \equiv 4^{392} \pmod{10}$
$$4^1 \equiv 4 \pmod{10} \\
4^2 \equiv 6  \\
4^3 \equiv 4 \\
4^4 \equiv 6 \\
(...)$$
Then I do $392 \pmod{2}$. Problem is that here, it's zero. If it were 1 or 2, the remainder would be either 4 or 6. How do I solve it this way?
 [/INST] $4^1\equiv 4$
$4^2\equiv 6$
$4^3\equiv 4$
$\vdots$
When the exponent is odd, the remainder is $4$.
When the exponent is even the remainder is $6$.
$0$ is even and so is $392$, thus $4^{392}\equiv 6\pmod{10}$

As an alternative notice that $6^k\equiv 6\pmod{10}$,
so $4^{392} = 16^{196}\equiv 6\pmod{10}$
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