what is the number positive integral solutions for xy = 72(x+y) ?


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Question Number 106409 by bemath last updated on 05/Aug/20

$what is the number positive integral solutions$ $for xy = 72 (x + y) ?$
	Answered by 1549442205PVT last updated on 05/Aug/20
	$We have xy=72(x+y)⇔(x−72)(y−72)=72^2 (∗) =(8.9)^2 =2^6 .3^4 .Hence (x−72)∣(2^6 .3^4 ) Since (2^6 .3^4 )has (6+1)(4+1)=35 positive divisors,the given equation has 35 poitive integer solutions Note 35 positive divisors of 72^2 =2^6 .3^4 are:{1,2,3,4,6, 8, 9, 12, 16, 18, 24, 27,32 48, 36,54, 64, 72, 81, 96, 108, 144, 162, 192,216,288 ,324,432, 576, 648, 864, 1296,1728,2592 ,5184}$
	$We have$ $xy = 72 (x + y) \Leftrightarrow (x - 72) (y - 72) = 72^{2} (*)$ $= {(8.9)}^{2} = 2^{6} . 3^{4} . Hence (x - 72) ∣ (2^{6} . 3^{4})$ $Since (2^{6} . 3^{4}) has (6 + 1) (4 + 1) = 35 positive$ $divisors, the given equation has 35$ $poitive integer solutions$ $Note 35 positive divisors of 72^{2} = 2^{6} . 3^{4}$ $are : {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32$ $48, 36, 54, 64, 72, 81, 96, 108, 144, 162,$ $192, 216, 288, 324, 432, 576, 648, 864,$ $1296, 1728, 2592, 5184}$
	Commented by Rasheed.Sindhi last updated on 05/Aug/20

	$xy = 72 (x + y) \Leftrightarrow (x - 72) (y - 72) = 72^{2} (*)$ $? ? ?$
	Commented by bemath last updated on 05/Aug/20

	$xy = 72 x + 72 y adding both sides by 72^{2}$ $xy + 72^{2} = 72 x + 72 y + 72^{2}$ $xy - 72 x - 72 y + 72^{2} = 72^{2}$ $x (y - 72) - 72 (y - 72) = 72^{2}$ $(y - 72) (x - 72) = 72^{2}$ $sir$
	Commented by Rasheed.Sindhi last updated on 05/Aug/20

	$T h a n k y o u b e m a t h s i r!$
	Commented by Rasheed.Sindhi last updated on 05/Aug/20

	$V . N i c e s i r 1549 . . .$
	Answered by Rasheed.Sindhi last updated on 05/Aug/20

	$W r o n g A p p r o a c h$ $\underline{S o l v e d f o r i n t e g r a l s o l u t i o n s .}$ $^{★} A s t h e e x p r e s s i o n i s s y m m e t r i c$ $s o i f (a, b) s a t i s f y, (b, a) a l s o s a t i s f y .$ $x y = 72 (x + y) \Rightarrow$ $x \times \frac{y}{x + y} = 72$ $(x & \frac{y}{x + y} a r e d i v i s o r s o f 72 a n d$ $p r o d u c t o f b o t h i s 72)$ $\| \begin{matrix} x & \frac{y}{x + y} & = \frac{y}{x + y} \\ 1 & 72 & = \frac{y}{1 + y} & y = \frac{- 72}{71} \notin Z \\ 2 & 36 & = \frac{y}{2 + y} & y = \frac{- 72}{35} \notin Z \\ 3 & 24 & = \frac{y}{3 + y} & y = \frac{- 72}{23} \notin Z \\ 4 & 18 & = \frac{y}{4 + y} & y = \frac{- 72}{17} \notin Z \\ 6 & 12 & = \frac{y}{6 + y} & y = \frac{- 72}{11} \notin Z \\ 8 & 9 & = \frac{y}{8 + y} & y = \frac{- 72}{8} = - 9 \\ 9 & 8 & = \frac{y}{9 + y} & y = \frac{- 72}{7} \notin Z \\ 12 & 6 & = \frac{y}{12 + y} & y = \frac{- 72}{5} \notin Z \\ 18 & 4 & = \frac{y}{18 + y} & y = \frac{- 72}{3} = - 24 \\ 24 & 3 & = \frac{y}{24 + y} & y = \frac{- 72}{2} = - 36 \\ 36 & 2 & = \frac{y}{36 + y} & y = \frac{- 72}{1} = - 72 \\ 72 & 1 & = \frac{y}{72 + y} & y = N o v a l u e \end{matrix} \|$ $(8, - 9), (- 9, 8), (18, - 24), (- 24, 18)$ $(24, - 36), (- 36, 24), (36, - 72), (- 72, 36)$ $8 i n t e g r a l s o l u t i o n s$ $A n s :$ $N o p o s i t i v e i n t e g r a l s o l u t i o n .$
	Commented by Her_Majesty last updated on 05/Aug/20

	$y o u a r e w r o n g$ $x y = 72 (x + y) \Rightarrow y = \frac{72 x}{x - 72}$ $i f w e l o o k f o r x ⩽ y a n d x, y > 0 \Rightarrow 72 < x ⩽ 144$ $x \in {73, 74, 75, 76, 78, 80, 81, 84, 88, 90, 96, 99, 104, 108, 120, 126, 136, 144}$ $\Rightarrow 35 s o l u t i o n s$ $y = \frac{72 x}{x - 72}$ $x = 72 + n$ $y = \frac{72 (n + 72)}{n} \Rightarrow n ∣ 72 \lor n ∣ (n + 72)$
	Commented by Rasheed.Sindhi last updated on 05/Aug/20

	$T h a n k s m a d a m! N o w I$ $r e c o g n i z e d m y l o g i c a l e r r o r .$ $T h e a n s w e r i s n o t e d i t a b l e b u t$ ${i t}^{'} s c o m p l e t e l y d i s c a r d a b l e!$
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