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Question Number 157332 by cortano last updated on 22/Oct/21

	Answered by Rasheed.Sindhi last updated on 22/Oct/21

	$6 x - y = 7 m \Rightarrow y = 6 x - 7 m$ $P (x) = 54 x^{2} + 3 x y - 2 y^{2} + z + 2021$ $= 54 x^{2} + 3 x (6 x - 7 m) - 2 {(6 x - 7 m)}^{2} + z + 2021$ $= 54 x^{2} + 18 x^{2} - 21 m x - 2 (36 x^{2} - 84 m x + 49 m^{2}) + z + 2021$ $= 54 x^{2} + 18 x^{2} - 21 m x - 72 x^{2} + 168 m x - 98 m^{2} + z + 2021$ $= 147 m x - 98 m^{2} + z + 2021$ $= 49 \times 3 m x - 49 \times 2 m^{2} + 49 \times 41 + 12 + z$ $= 49 (3 m x - 2 m^{2} + 41) + 12 + z$ $49 ∣ p (x)$ $\Rightarrow 49 ∣ (12 + z)$ $\Rightarrow z + 12 \equiv 0 (m o d 49)$ $z \equiv - 12 + 49 (m o d 49$ $z \equiv 37 (m o d 49)$ $z = 37 + 49 k$ $M i n i m u m p o s i t i v e o f z = 37 + 49 \times 0$ $z = 37$
	Commented by cortano last updated on 22/Oct/21

	$t h a n k y o u . i t g r e a t w a y$
	Commented by mr W last updated on 22/Oct/21

	$i t h i n k w i t h t h e c o n d i t i o n s g i v e n i n$ $t h e q u e s t i o n t h e r e i s n o r e q u e s t e d$ $s o l u t i o n p o s s i b l e .$ $h e r e j u s t a n e x a m p l e w h i c h a l s o$ $f u l f i l l s t h e c o n d i t i o n s :$ $x = \frac{1}{7}$ $y = - \frac{43}{7}$ $z = 16 < 37$
	Commented by Rasheed.Sindhi last updated on 22/Oct/21

	${You}^{'} re very right sir! ActuallyI assumed x$ $x and y positive integers . And this$ $I did unconsciously . Anyway my solution$ $is only right for positive integers and$ $according to the question {you}^{'} re$ $right . It seems that the condition of$ $being integer of x and y is ignored$ $mistakenly .$
	Commented by mr W last updated on 22/Oct/21

	$i t h i n k s o t o o . x, y s h o u l d b e i n t e g e r .$
	Commented by Rasheed.Sindhi last updated on 22/Oct/21

	$Salute to your fine observation!$
	Answered by ajfour last updated on 22/Oct/21

	$6 x - y = 7 m; m \in Z$ $- z = 54 y^{2} (t^{2} + \frac{t}{18} - \frac{1}{27}) + p$ $f = 49 N = 54 (x - \frac{2 y}{9}) (x + \frac{y}{6}) + z + p$ $N = \frac{54 (x - \frac{2 y}{9}) (x + \frac{y}{6}) + (z + p)}{7}$ $28 N = (42 m - 2 y) (7 m + 2 y)$ $+ 2 (z + p)$ $\Rightarrow z i s a + m i n w h e n 2 (z + p) =$ $28 N + (2 y + 7 m) (2 y - 42 m)$ $i s a + m i n .$ $\Rightarrow 2 z = - 4042 + 28 N - \frac{63 \times 35 m^{2}}{4}$ $z = \frac{7 (16 N - 315 m^{2})}{8} - 2021$ $h o w t o o b t a i n a p o s i t i v e m i n$ $o u t f r o m t h i s, i j u s t d u n n o w . .$
	Commented by cortano last updated on 22/Oct/21

	$t h a n k y o u$
	Commented by mr W last updated on 22/Oct/21

	$a j f o u r s i r i s r i g h t . t h e r e i s n o u n i q u e$ $s o l u t i o n .$
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