Determine ab^(−) (a>b) such that( ab^(−) +ba^(−) ) and (ab^(−) −ba^(−) ) are both perfect squares.


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Question Number 203146 by Rasheed.Sindhi last updated on 11/Jan/24

$D e t e r m i n e \overset{―}{a b} (a > b) s u c h t h a t (\overset{―}{a b} + \overset{―}{b a})$ $a n d (\overset{―}{a b} - \overset{―}{b a}) a r e b o t h p e r f e c t s q u a r e s .$
	Answered by Frix last updated on 11/Jan/24

	$10 a + b + 10 b + a = p^{2} \Leftrightarrow 11 (a + b) = p^{2}$ $10 a + b - 10 b - a = q^{2} \Leftrightarrow 9 (a - b) = q^{2}$ $\Rightarrow$ $a + b = 11 \land (a - b = 1 \lor a - b = 4)$ $b = 11 - a \Rightarrow a - b = 2 a - 11$ $2 a - 11 = 1 \Rightarrow a = 6 \land b = 5 ★$ $2 a - 11 = 4 \Rightarrow a \notin N$
	Commented by Rasheed.Sindhi last updated on 11/Jan/24

	$T han k s sir!$
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