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?

Since you found $572 \equiv 2\pmod{10} $ you have to check only what is $$2^{42}  \equiv ?\pmod{10} $$
Since $2^5 \equiv 2$ we have $$2^{42} = 2^2\cdot (\color{red}{2^5})^8 \equiv 2^2\cdot \color{red}{2}^8 \equiv 2^{10} \equiv (\color{blue}{2^5})^2 \equiv \color{blue}{2}^2\equiv 4 \pmod {10}$$
