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?

Applying $\ xy\bmod xz\,  =\, x(y\bmod z)\ = $ mod Distributive Law to factor out $\,x = 2\,$ below
$\ \ \ \ \begin{align}(2a)^{\large 2+4N}\!\bmod 10 &\,=\, 2(2\, \color{#c00}2^{\large\color{#c00} 4N} a^{\large 2} \color{#c00}a^{\large\color{#c00} 4N}\bmod 5)\\[.2em]
&\,=\, 2(2a^{\large 2}\bmod 5)\ \ {\rm by}\ \ 5\nmid a\,\Rightarrow\ \color{#c00}{a^{\large 4}\equiv 1}\!\!\!\pmod{5}\\[.3em]
&\,=\, 4\ \ \ {\rm if}\ a\equiv \pm1\!\!\!\pmod{\!5}\end{align}$
