<s>[INST] Find the units digit of $572^{42}$ The idea of this exercise is that you use the modulus to get the right answer.
What I did was:
$$572\equiv 2\pmod {10} \\
572^2 \equiv 2^2  \equiv 4\pmod{10} \\
572^3 \equiv 2^3  \equiv 8\pmod{10} \\
572^4 \equiv 2^4  \equiv 6\pmod{10} \\
572^5 \equiv 2^5  \equiv 2\pmod{10} \\
572^6 \equiv 2^6  \equiv 4\pmod{10} \\
(...)$$
I can see that this goes 2,4,8,6 and then repeats. I remember that the gist of the exercise is to find the remainder based on this repetition. How do I do that? I know that $572^{42} \equiv 2^{42}\equiv ? \pmod {10}$. How do I simplify that 42 and answer this using that repetition?
 [/INST] Another approach is:
The reminder is repeating between:
$$R=\{2,4,8,6\}$$
Solve the following, since $|R|=4$
$$42\equiv x\pmod{4}$$
Have $x=2$, which means the reminder is the second term in $R$ that is $4$
Therefore $$572^{42}\equiv4\pmod{10}$$
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