Prove by maths induction tbat n^5 − n^3 is divisible by 24.


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Question Number 82034 by Dah Solu Tion last updated on 17/Feb/20

$P r o v e b y m a t h s i n d u c t i o n t b a t$ $n^{5} - n^{3} i s d i v i s i b l e b y 24 .$
Commented by MJS last updated on 17/Feb/20
$without induction 24=2^3 ×3 n^5 −n^3 =(n−1)n^3 (n+1) ∀n∈Z: 3∣(n−1) xor 3∣n xor 3∣(n+1) ⇒ 3∣(n^5 −n^3 ) ∀n∈Z: 2∣(n−1)∧2∣(n+1) xor 2∣n ⇒ { ((2∣n ⇒ 8∣n^3 )),((2∤n ⇒ n=2k+1 ⇒ (n−1)(n+1)=4k(k+1) ⇒ 8∣(n−1)(n+1))) :} ⇒ 8∣(n^5 −n^3 ) ⇒ 24∣(n^5 −n^3 )$
$without induction 24 = 2^{3} \times 3$ $n^{5} - n^{3} = (n - 1) n^{3} (n + 1)$ $\forall n \in Z : 3 ∣ (n - 1) xor 3 ∣ n xor 3 ∣ (n + 1)$ $\Rightarrow 3 ∣ (n^{5} - n^{3})$ $\forall n \in Z : 2 ∣ (n - 1) \land 2 ∣ (n + 1) xor 2 ∣ n$ $\Rightarrow {\begin{cases} 2 ∣ n \Rightarrow 8 ∣ n^{3} \\ 2 ∤ n \Rightarrow n = 2 k + 1 \Rightarrow (n - 1) (n + 1) = 4 k (k + 1) \Rightarrow 8 ∣ (n - 1) (n + 1) \end{cases}$ $\Rightarrow 8 ∣ (n^{5} - n^{3})$ $\Rightarrow 24 ∣ (n^{5} - n^{3})$
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