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Question Number 110592 by Aina Samuel Temidayo last updated on 29/Aug/20

$$\mathrm{How}\mathrm{many}\mathrm{pairs}\mathrm{of}\mathrm{integers}\mathrm{x}\mathrm{and}\mathrm{y}$$$$\mathrm{satisfy}\mathrm{the}\mathrm{equation}\frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}=\frac{1}{32}$$

Answered by Aziztisffola last updated on 29/Aug/20

$$\mathrm{if}\mathrm{x}=\mathrm{y}\Rightarrow \frac{2}{\mathrm{x}}=\frac{1}{32}\Rightarrow \mathrm{x}=64$$$$(\mathrm{x};\mathrm{y})=(64;64)\in \mathrm{S}\mathrm{then}\mathrm{S}\ne \varnothing$$$$\mathrm{there}\mathrm{is}\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{solution}.$$

Answered by mr W last updated on 29/Aug/20

$$\frac{1}{y}=\frac{1}{32}-\frac{1}{x}=\frac{x-32}{32x}$$$$y=\frac{32x}{x-32}$$$$x=33\Rightarrow y=1056$$$$x=34\Rightarrow y=544$$$$x=36\Rightarrow y=288$$$$x=40\Rightarrow y=160$$$$x=48\Rightarrow y=96$$$$x=64\Rightarrow y=64$$$$x=31\Rightarrow y=-993$$$$x=30\Rightarrow y=-480$$$$x=28\Rightarrow y=-224$$$$x=24\Rightarrow y=-96$$$$x=16\Rightarrow y=-32$$$$......$$

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

$$\mathrm{Thanks}\mathrm{but}\mathrm{how}\mathrm{did}\mathrm{you}\mathrm{know}\mathrm{the}\mathrm{values}\mathrm{of}(\mathrm{x},\mathrm{y})$$$$\mathrm{that}\mathrm{fits}\mathrm{the}\mathrm{equation}?\mathrm{Your}\mathrm{work}\mathrm{is}$$$$\mathrm{not}\mathrm{well}\mathrm{explanatory}.$$

Commented by mr W last updated on 29/Aug/20

$$ifx-32divides32,thenxisok.$$$$\Rightarrow x-32=\pm 1,\pm 2,\pm 4,\pm 8,\pm 16,\pm 32$$

Answered by floor(10²Eta[1]) last updated on 29/Aug/20

$$\mathrm{go}\mathrm{see}\mathrm{my}\mathrm{answer}\mathrm{on}\mathrm{Q}.103093$$$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{the}\mathrm{same}\mathrm{logic}$$

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

$$\mathrm{Thanks}.$$

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