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Question Number 192425 by mehdee42 last updated on 17/May/23

Question if ♮kε is odd & A=1^k +2^k +...+n^(k ) & B=1+2+...+n prove that : B ∣ A

$${Question} \\ $$ $${if}\:\:\natural{k}\varepsilon\:{is}\:{odd}\:\:\&\:{A}=\mathrm{1}^{{k}} +\mathrm{2}^{{k}} +...+{n}^{{k}\:\:} \:\&\:\:{B}=\mathrm{1}+\mathrm{2}+...+{n} \\ $$ $${prove}\:{that}\:\::\:\:{B}\:\mid\:{A}\: \\ $$

Answered by MM42 last updated on 24/May/23

We khnow ; 1+2+...+n=((n(n+1))/2) & (n,((n+1)/2))=1 or ((n/2),n+1)=1 Thus ,it soffices to show that n∣A & ((n+1)/2) ∣A or (n/2)∣A & n+1∣A proof ∴ if ♮nε is even ⇒ (n/2)∣1^k +(n−1)^k , 2^k +(n−2)^k , ... (i) n+1∣1^k +n^k , 2^k +(n−1)^k , ... (ii) (i),(ii)⇒B∣A if ♮nε is odd⇒ n∣1^k +(n−1)^k , 2^k +(n−2)^k , ... (iii) ((n+1)/2)∣1^k +n^k , 2^k +(n−1)^k , ... (iv) (iii),(iv)⇒B∣A ⇒∀ n∈N ⇒1+2+3+...+n∣ 1^k +2^k +...+n^k ; if ♮kε is ♮oddε

$${We}\:{khnow}\:\:;\:\:\mathrm{1}+\mathrm{2}+...+{n}=\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:\:\:\&\:\:\:\left({n},\frac{{n}+\mathrm{1}}{\mathrm{2}}\right)=\mathrm{1}\:{or}\:\left(\frac{{n}}{\mathrm{2}},{n}+\mathrm{1}\right)=\mathrm{1} \\ $$ $${Thus}\:,{it}\:{soffices}\:\:{to}\:\:{show}\:{that}\: \\ $$ $${n}\mid{A}\:\&\:\frac{{n}+\mathrm{1}}{\mathrm{2}}\:\mid{A}\:\:{or}\:\:\frac{{n}}{\mathrm{2}}\mid{A}\:\&\:{n}+\mathrm{1}\mid{A} \\ $$ $${proof}\:\therefore \\ $$ $${if}\:\:\natural{n}\varepsilon\:{is}\:{even}\:\Rightarrow \\ $$ $$\:\frac{{n}}{\mathrm{2}}\mid\mathrm{1}^{{k}} +\left({n}−\mathrm{1}\right)^{{k}} \:,\:\mathrm{2}^{{k}} +\left({n}−\mathrm{2}\right)^{{k}} \:,\:...\:\left({i}\right) \\ $$ $$\:{n}+\mathrm{1}\mid\mathrm{1}^{{k}} +{n}^{{k}} \:,\:\mathrm{2}^{{k}} +\left({n}−\mathrm{1}\right)^{{k}} \:,\:...\:\left({ii}\right) \\ $$ $$\left({i}\right),\left({ii}\right)\Rightarrow{B}\mid{A} \\ $$ $${if}\:\:\natural{n}\varepsilon\:{is}\:\:{odd}\Rightarrow \\ $$ $$\:{n}\mid\mathrm{1}^{{k}} +\left({n}−\mathrm{1}\right)^{{k}} \:,\:\mathrm{2}^{{k}} +\left({n}−\mathrm{2}\right)^{{k}} \:,\:...\:\left({iii}\right) \\ $$ $$\:\frac{{n}+\mathrm{1}}{\mathrm{2}}\mid\mathrm{1}^{{k}} +{n}^{{k}} \:,\:\mathrm{2}^{{k}} +\left({n}−\mathrm{1}\right)^{{k}} \:,\:...\:\left({iv}\right) \\ $$ $$\left({iii}\right),\left({iv}\right)\Rightarrow{B}\mid{A} \\ $$ $$\Rightarrow\forall\:{n}\in\mathbb{N}\:\Rightarrow\mathrm{1}+\mathrm{2}+\mathrm{3}+...+{n}\mid\:\mathrm{1}^{{k}} +\mathrm{2}^{{k}} +...+{n}^{{k}} \:\:\:;\:{if}\:\:\natural{k}\varepsilon\:{is}\:\:\natural{odd}\varepsilon \\ $$ $$ \\ $$

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