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Question Number 53770 by Tawa1 last updated on 25/Jan/19

$$\mathrm{Solve}\mathrm{the}\mathrm{equation}:$$$${\left(\mathrm{he}\right)}^2=\mathrm{she},\mathrm{where}\mathrm{h},\mathrm{e}\mathrm{and}\mathrm{s}\mathrm{are}\mathrm{integers}.$$

Answered by mr W last updated on 26/Jan/19

$$lethe=t$$$$t^2=100s+t$$$$t^2-t-100s=0$$$$t=\frac{1+\sqrt{1+400s}}{2}$$$$1+400s={(2n+1)}^2$$$$2n\times (2n+2)=400s$$$$\Rightarrow n(n+1)=100s$$$$thereisonlyonepossibility:$$$$24\times 25=100\times 6$$$$i.e.s=6,n=24$$$$\Rightarrow t=\frac{1+49}{2}=25=he$$$$\Rightarrow {\left(25\right)}^2=625$$

Commented by peter frank last updated on 26/Jan/19

$$sirhow{t}^2=100s+t$$

Commented by Tawa1 last updated on 26/Jan/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Commented by $@ty@m last updated on 26/Jan/19

$$\because \mathrm{she}isa3digitnumber.$$$$\therefore \mathrm{she}=100\mathrm{s}+10\mathrm{h}+\mathrm{e}$$$$\Rightarrow \mathrm{she}=100\mathrm{s}+\mathrm{he}$$$$\Rightarrow \mathrm{she}=100\mathrm{s}+\mathrm{t}(\because \mathrm{t}=\mathrm{he})$$

Commented by peter frank last updated on 27/Jan/19

$$thanks$$

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