Power of 2^10 Minus Power of 10^3 is Divisible by 24

Theorem
Let $x \in \Z_{\ge 0}$ be a non-negative integer.

Then $2^{10 x} - 10^{3 x}$ is divisible by $3$.

That is:
 * $2^{10 x} - 10^{3 x} \equiv 0 \pmod 3$

Examples
Let us examine arguably the flagship example, the smallest number whose compliance is not obvious nor trivial:
 * $1 \, 048 \, 576$

This is equal to $2^{20}$, which is equal to $2^{10 \times 2}$, thus one of the valid powers of $2$.

We then subtract $10^{3 \times 2}$, or $10^6$:


 * $1 \, 048 \, 576 - 1 \, 000 \, 000 = 48 \, 576$

and we see that:
 * $48 \, 576 = 16 \, 192 \times 3$

satisfying the theorem.