a, b∈N_+ , ((b+1)/a)+((a+1)/b)∈Z. Prove that (a, b)≤(√(a+b.))


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Question Number 207436 by CrispyXYZ last updated on 15/May/24

$a, b \in N_{+}, \frac{b + 1}{a} + \frac{a + 1}{b} \in Z . Prove that$ $(a, b) ⩽ \sqrt{a + b .}$
Commented by A5T last updated on 15/May/24

$W h a t i s m e a n t b y (a, b) ?$
Commented by CrispyXYZ last updated on 15/May/24

$the greatest common divisor of a, b$
	Answered by Berbere last updated on 16/May/24

	$\frac{b^{2} + a^{2} + a + b}{a b} \in Z$ $a b ∣ {(a + b)}^{2} - a b + (a + b); a ∣ b \Leftrightarrow a = k b$ $\Rightarrow a b ∣ (a + b) {(1 + (a + b))}^{'} . . (E)$ $l e t d = (a, b) \Rightarrow a = {d a}^{'}; b = {d b}^{'}$ $E \Leftrightarrow d^{2} a^{'} b^{'} ∣ ({d a}^{'} + {d b}^{'}) (1 + d (a^{'} + b^{'}))$ $d^{2} ∣ d^{2} a^{'} b^{'} \Rightarrow d^{2} ∣ ({d a}^{'} + {d b}^{'}) + d^{2} {(a^{'} + b^{'})}^{2}$ $\Rightarrow d^{2} ∣ d (a^{'} + b^{'}) \Leftrightarrow d^{2} ∣ {d a}^{'} + {d b}^{'} \Leftrightarrow d^{2} ∣ a + b$ $\Rightarrow d^{2} ⩽ a + b \Rightarrow d ⩽ \sqrt{a + b}$
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