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?

You want $2^{42} \mod 10$.  You found that $2^4\equiv6\mod 10$.  Now $6^n\equiv 6$ for all $n\in\mathbb N$ (you could prove that by induction), so $2^{42}\equiv2^{40}2^2\equiv(2^4)^{10}2^2\equiv6^{10}\times4\equiv6\times4\equiv4\mod10$.  
