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Question Number 192786 by MM42 last updated on 27/May/23

$$N=<aabb>\in \mathbb{N}\mathrm{\&}Nisperfectsquare$$ $$findN?$$ $$$$

Answered by AST last updated on 27/May/23

$$1100a+11b=x^2\Rightarrow 11(100a+b)=x^2$$ $$11\mid 100a+b\Rightarrow 100a+b=11p^2\Rightarrow 11\mid a+b$$ $$(a,b)\ne (6,5),b\ne 2,3,6,7,8(aperfectsquarecannot$$ $$endin2,3,7or8,powerof2in\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}{aabb}whenb=6is1)$$ $$\Rightarrow Possiblevaluesof(a,b)=(2,9),(7,4)$$ $$Ofthese,only(7,4)givesaperfectsquare$$ $$\Rightarrow N=7744$$

Commented byBaliramKumar last updated on 27/May/23

$$\mathrm{perfect}\mathrm{square}\mathrm{number}\mathrm{last}\mathrm{two}\mathrm{digit}\mathrm{only}$$ $$00\mathrm{or}44$$ $$\because 11\mid \mathrm{a}+\mathrm{b}$$ $$\therefore \mathrm{bb}\ne 00$$ $$(\mathrm{if}\mathrm{bb}=00\mathrm{then}11=\mathrm{a}>9,{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{not}\mathrm{possible})$$

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