Given 2x^2 y^2 +12y^2 =7x^2 +647 for x,y ε Z . Find the remaider if 3x^2 y^4 divide by 11 .


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Question Number 178686 by greougoury555 last updated on 20/Oct/22

$G i v e n 2 x^{2} y^{2} + 12 y^{2} = 7 x^{2} + 647$ $f o r x, y ε Z .$ $F i n d t h e r e m a i d e r i f 3 x^{2} y^{4} d i v i d e b y$ $11 .$
	Answered by Rasheed.Sindhi last updated on 20/Oct/22

	$2 x^{2} y^{2} + 12 y^{2} = 7 x^{2} + 647$ $2 y^{2} (x^{2} + 6) = 7 (x^{2} + 6) + 605$ $2 y^{2} (x^{2} + 6) - 7 (x^{2} + 6) = 605$ $(x^{2} + 6) (2 y^{2} - 7) = 5 \times 11^{2} . . . . . . A$ $P o s s i b l e v a l u e s f o r x^{2} + 6 :$ $1, 5, 11, 121, 55, 605, - 1, - 5, - 11, - 121, - 55, - 605$ $P o s s i b l e v a l u e s f o r x^{2}$ $\overset{\times}{- 5}, \overset{\times}{- 1}, \overset{\times}{5}, \overset{\times}{115}, {\begin{matrix} 49 \end{matrix}}^{✓}, \overset{\times}{599}, \overset{\times}{- 7}, \overset{\times}{- 11}, \overset{\times}{- 17}, \overset{\times}{- 127}, \overset{\times}{- 61}, \overset{\times}{- 611}$ $∵ x \in Z ∴ x^{2} = 49 \Rightarrow x = \pm 7$ $I f x^{2} + 6 = {(\pm 7)}^{2} + 6 = 55 t h e n f r o m A$ $2 y^{2} - 7 = 605 / 55 = 11$ $2 y^{2} = 18 \Rightarrow y = \pm 3$ $(x, y) = (\pm 7, \pm 3) o r (\pm 7, \mp 3)$ $3 x^{2} y^{4} = 3 {(\pm 7)}^{2} {(\pm 3)}^{4} = 3 {(\pm 7)}^{2} {(\mp 3)}^{4} = 11907$ $11907 (m o d 11) = \begin{matrix} 5 \end{matrix}$
	Commented by cortano1 last updated on 21/Oct/22

	$typo \Rightarrow {2 y}^{2} = 18; y = \pm 3$ $(x, y) = (\pm 7, \pm 3)$ $then {3 x}^{2} y^{4} = 3 \times 49 \times 81 = 11907$ $11907 = 1082 \times 11 + 5 = 5 (\mod 11)$
	Commented by Rasheed.Sindhi last updated on 20/Oct/22

	$T h a n k y o u s i r, I^{'} v e c o r r e c t e d n o w .$
	Answered by greougoury555 last updated on 21/Oct/22
	$2x^2 y^2 −7x^2 = 647−12y^2 x^2 (2y^2 −7)=605+42−12y^2 x^2 (2y^2 −7)=605+6(7−2y^(2)) x^2 (2y^2 −7)+6(2y^2 −7)=605 (2y^2 −7)(x^2 +6)=11×55 { ((2y^2 −7=11⇒y^2 =9)),((x^2 +6=55⇒x=49)) :} ⇒3x^2 y^4 = 3×49×81=11097 ⇒((11097)/(11)) = 1082+(5/(11))$
	$2 x^{2} y^{2} - 7 x^{2} = 647 - 12 y^{2}$ $x^{2} (2 y^{2} - 7) = 605 + 42 - 12 y^{2}$ $x^{2} (2 y^{2} - 7) = 605 + 6 (7 - 2 y^{2)}$ $x^{2} (2 y^{2} - 7) + 6 (2 y^{2} - 7) = 605$ $(2 y^{2} - 7) (x^{2} + 6) = 11 \times 55$ ${\begin{cases} 2 y^{2} - 7 = 11 \Rightarrow y^{2} = 9 \\ x^{2} + 6 = 55 \Rightarrow x = 49 \end{cases}$ $\Rightarrow 3 x^{2} y^{4} = 3 \times 49 \times 81 = 11097$ $\Rightarrow \frac{11097}{11} = 1082 + \frac{5}{11}$
	Commented by Rasheed.Sindhi last updated on 21/Oct/22

	$T y p o : 3 \times 49 \times 81 = 11907$
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