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Question-151265




Question Number 151265 by liberty last updated on 19/Aug/21
Commented by Tawa11 last updated on 19/Aug/21
Weldone sir.
$$\mathrm{Weldone}\:\mathrm{sir}. \\ $$
Commented by liberty last updated on 19/Aug/21
5x ≡ 2 (mod 3)⇒x=1+3a  3x ≡ 4 (mod 7) ⇒9a+3≡4 (mod 7)  ⇒2a ≡ 1 (mod 7) ⇒a = 4+7b  3x ≡ 7 (mod 8)⇒3(1+3(4+7b))≡ 7(mod 8)  ⇒3(13+21b)≡ 7 (mod 8)  ⇒63b+39 ≡ 7 (mod 8)  ⇒7b+7 ≡ 7 (mod 8)  ⇒7b ≡ 0 (mod 8) ⇒b ≡ 0 (mod 8)  ⇒b = 8c   then x = 1+3a = 1+3(4+7b)  x=21b+13 = 21(8c)+13  x=168c +13   the smallest positive integer   x = 13 .
$$\mathrm{5x}\:\equiv\:\mathrm{2}\:\left(\mathrm{mod}\:\mathrm{3}\right)\Rightarrow\mathrm{x}=\mathrm{1}+\mathrm{3a} \\ $$$$\mathrm{3x}\:\equiv\:\mathrm{4}\:\left(\mathrm{mod}\:\mathrm{7}\right)\:\Rightarrow\mathrm{9a}+\mathrm{3}\equiv\mathrm{4}\:\left(\mathrm{mod}\:\mathrm{7}\right) \\ $$$$\Rightarrow\mathrm{2a}\:\equiv\:\mathrm{1}\:\left(\mathrm{mod}\:\mathrm{7}\right)\:\Rightarrow\mathrm{a}\:=\:\mathrm{4}+\mathrm{7b} \\ $$$$\mathrm{3x}\:\equiv\:\mathrm{7}\:\left(\mathrm{mod}\:\mathrm{8}\right)\Rightarrow\mathrm{3}\left(\mathrm{1}+\mathrm{3}\left(\mathrm{4}+\mathrm{7b}\right)\right)\equiv\:\mathrm{7}\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$$$\Rightarrow\mathrm{3}\left(\mathrm{13}+\mathrm{21b}\right)\equiv\:\mathrm{7}\:\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$$$\Rightarrow\mathrm{63b}+\mathrm{39}\:\equiv\:\mathrm{7}\:\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$$$\Rightarrow\mathrm{7b}+\mathrm{7}\:\equiv\:\mathrm{7}\:\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$$$\Rightarrow\mathrm{7b}\:\equiv\:\mathrm{0}\:\left(\mathrm{mod}\:\mathrm{8}\right)\:\Rightarrow\mathrm{b}\:\equiv\:\mathrm{0}\:\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$$$\Rightarrow\mathrm{b}\:=\:\mathrm{8c}\: \\ $$$$\mathrm{then}\:\mathrm{x}\:=\:\mathrm{1}+\mathrm{3a}\:=\:\mathrm{1}+\mathrm{3}\left(\mathrm{4}+\mathrm{7b}\right) \\ $$$$\mathrm{x}=\mathrm{21b}+\mathrm{13}\:=\:\mathrm{21}\left(\mathrm{8c}\right)+\mathrm{13} \\ $$$$\mathrm{x}=\mathrm{168c}\:+\mathrm{13}\: \\ $$$$\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{integer}\: \\ $$$$\mathrm{x}\:=\:\mathrm{13}\:.\: \\ $$
Commented by otchereabdullai@gmail.com last updated on 22/Aug/21
nice
$$\mathrm{nice} \\ $$
Answered by Rasheed.Sindhi last updated on 19/Aug/21
5x≡2+3=5(mod 3)  x≡1(mod 3)..................(i)  3x≡4(mod 7  3x≡4+7×2=18(mod 7  x≡6(mod7)....................(ii)  3x≡7(mod 8)  3x≡7+8=15(mod 8)  x≡5(mod8)....................(iii)  (i)⇒x=1,4,7,10,13,16,19,....  (ii)⇒6,13,20,27,....  (iii)5,13,21,29,...  x=13
$$\mathrm{5}{x}\equiv\mathrm{2}+\mathrm{3}=\mathrm{5}\left({mod}\:\mathrm{3}\right) \\ $$$${x}\equiv\mathrm{1}\left({mod}\:\mathrm{3}\right)………………\left({i}\right) \\ $$$$\mathrm{3}{x}\equiv\mathrm{4}\left({mod}\:\mathrm{7}\right. \\ $$$$\mathrm{3}{x}\equiv\mathrm{4}+\mathrm{7}×\mathrm{2}=\mathrm{18}\left({mod}\:\mathrm{7}\right. \\ $$$${x}\equiv\mathrm{6}\left({mod}\mathrm{7}\right)………………..\left({ii}\right) \\ $$$$\mathrm{3}{x}\equiv\mathrm{7}\left({mod}\:\mathrm{8}\right) \\ $$$$\mathrm{3}{x}\equiv\mathrm{7}+\mathrm{8}=\mathrm{15}\left({mod}\:\mathrm{8}\right) \\ $$$${x}\equiv\mathrm{5}\left({mod}\mathrm{8}\right)………………..\left({iii}\right) \\ $$$$\left({i}\right)\Rightarrow{x}=\mathrm{1},\mathrm{4},\mathrm{7},\mathrm{10},\mathrm{13},\mathrm{16},\mathrm{19},…. \\ $$$$\left({ii}\right)\Rightarrow\mathrm{6},\mathrm{13},\mathrm{20},\mathrm{27},…. \\ $$$$\left({iii}\right)\mathrm{5},\mathrm{13},\mathrm{21},\mathrm{29},… \\ $$$${x}=\mathrm{13} \\ $$
Commented by liberty last updated on 19/Aug/21
okay. thanks
$$\mathrm{okay}.\:\mathrm{thanks} \\ $$

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