Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 128636 by john_santu last updated on 09/Jan/21

$$\mathrm{Solve}\mathrm{diopthantine}\mathrm{equation}$$$$\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}=\frac{2}{17}.$$

Answered by liberty last updated on 09/Jan/21

$$\Rightarrow \frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}=\frac{2}{17};2\mathrm{ab}=17(\mathrm{a}+\mathrm{b})$$$$\mathrm{consider}(2\mathrm{a}-17)(2\mathrm{b}-17)=4\mathrm{ab}-34(\mathrm{a}+\mathrm{b})+{17}^2$$$$\mathrm{it}\mathrm{follows}\mathrm{that}4\mathrm{ab}-34(\mathrm{a}+\mathrm{b})=0$$$$\mathrm{so}\mathrm{we}\mathrm{get}(2\mathrm{a}-17)(2\mathrm{b}-17)={17}^2$$$$\mathrm{since}\mathrm{a}\mathrm{and}\mathrm{b}\mathrm{are}\mathrm{positive}\mathrm{integer}\mathrm{we}\mathrm{can}$$$$\mathrm{take}\left(\mathrm{i}\right)2\mathrm{a}-17=1\mathrm{and}2\mathrm{a}-17={17}^2$$$$\mathrm{which}\mathrm{yields}(\mathrm{a},\mathrm{b})=(9,103)$$$$\left(\mathrm{ii}\right)2\mathrm{a}-17=17\mathrm{and}2\mathrm{a}-17=17\mathrm{which}\mathrm{yields}$$$$(\mathrm{a},\mathrm{b})=(17,17)$$$$\left(\mathrm{iii}\right)2\mathrm{a}-17={17}^2\mathrm{and}2\mathrm{a}-17=1\mathrm{which}\mathrm{yields}$$$$(\mathrm{a},\mathrm{b})=(103,9)$$$$\therefore \equiv \mathrm{solution}\{(9,103),(17,17),(103,9)\}$$$$$$$$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com