Power of 2^10 Minus Power of 10^3 is Divisible by 24/Proof 3
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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:
- $2^{10 n} - 10^{3 n} \equiv 0 \pmod {24}$
Proof
\(\ds 2^{10 n} - 10^{3 n}\) | \(=\) | \(\ds \paren {2^{10} }^n - \paren {10^3}^n\) | Power of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2^{10} - 10^3} \sum_{j \mathop = 0}^{n - 1} \paren {2^{10} }^{n - j - 1} \paren {10^3}^j\) | Difference of Two Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds 24 k\) | where $\ds k = \sum_{j \mathop = 0}^{n - 1} {2^{10} }^{n - j - 1} \paren {10^3}^j$ is an integer | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2^{10 n} - 10^{3 n}\) | \(\equiv\) | \(\ds 0 \pmod {24}\) | Definition of Congruence Modulo Integer |
$\blacksquare$