Prove-by-maths-induction-tbat-n-5-n-3-is-divisible-by-24-

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} \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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