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

Theorem

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

Then $2^{10 n} - 10^{3 n}$ is divisible by $24$.

That is:

$2^{10 n} - 10^{3 n} \equiv 0 \pmod {24}$

$\ds 2^{10 n} - 10^{3 n}$

$=$

$\ds \paren {2^{10} }^n - \paren {10^3}^n$

$\ds $

$=$

$\ds \paren {2^{10} - 10^3} \sum_{j \mathop = 0}^{n - 1} \paren {2^{10} }^{n - j - 1} \paren {10^3}^j$

$\ds $

$=$

$\ds 24 k$

where $\ds k = \sum_{j \mathop = 0}^{n - 1} {2^{10} }^{n - j - 1} \paren {10^3}^j$ is an integer

$\ds \leadsto \ \ $

$\ds 2^{10 n} - 10^{3 n}$

$\equiv$

$\ds 0 \pmod {24}$

$\blacksquare$