Question Number 53770 by Tawa1 last updated on 25/Jan/19
$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{he}\right)^{\mathrm{2}} \:\:=\:\:\mathrm{she}\:\:,\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{where}\:\:\mathrm{h},\:\mathrm{e}\:\:\mathrm{and}\:\:\mathrm{s}\:\:\mathrm{are}\:\mathrm{integers}\:. \\ $$
Answered by mr W last updated on 26/Jan/19
$${let}\:{he}={t} \\ $$$${t}^{\mathrm{2}} =\mathrm{100}{s}+{t} \\ $$$${t}^{\mathrm{2}} −{t}−\mathrm{100}{s}=\mathrm{0} \\ $$$${t}=\frac{\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{400}{s}}}{\mathrm{2}} \\ $$$$\mathrm{1}+\mathrm{400}{s}=\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\mathrm{2}{n}×\left(\mathrm{2}{n}+\mathrm{2}\right)=\mathrm{400}{s} \\ $$$$\Rightarrow{n}\left({n}+\mathrm{1}\right)=\mathrm{100}{s} \\ $$$${there}\:{is}\:{only}\:{one}\:{possibility}: \\ $$$$\mathrm{24}×\mathrm{25}=\mathrm{100}×\mathrm{6} \\ $$$${i}.{e}.\:{s}=\mathrm{6},\:{n}=\mathrm{24} \\ $$$$\Rightarrow{t}=\frac{\mathrm{1}+\mathrm{49}}{\mathrm{2}}=\mathrm{25}={he} \\ $$$$\Rightarrow\left(\mathrm{25}\right)^{\mathrm{2}} =\mathrm{625} \\ $$
Commented by peter frank last updated on 26/Jan/19
$${sir}\:{how}\:\:\:{t}^{\mathrm{2}} =\mathrm{100}{s}+{t} \\ $$
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
$$\because\mathrm{she}\:{is}\:{a}\:\mathrm{3}\:{digit}\:{number}. \\ $$$$\therefore\mathrm{she}=\mathrm{100s}+\mathrm{10h}+\mathrm{e} \\ $$$$\Rightarrow\mathrm{she}=\mathrm{100s}+\mathrm{he} \\ $$$$\Rightarrow\mathrm{she}=\mathrm{100s}+\mathrm{t}\:\left(\because\mathrm{t}=\mathrm{he}\right) \\ $$
Commented by peter frank last updated on 27/Jan/19
$${thanks} \\ $$