Question Number 192786 by MM42 last updated on 27/May/23
$${N}=<{aabb}>\in\mathbb{N}\:\:\&\:\:{N}\:\:{is}\:\:{perfect}\:{square} \\ $$$${find}\:\:{N}\:\:? \\ $$$$ \\ $$
Answered by AST last updated on 27/May/23
$$\mathrm{1100}{a}+\mathrm{11}{b}={x}^{\mathrm{2}} \Rightarrow\mathrm{11}\left(\mathrm{100}{a}+{b}\right)={x}^{\mathrm{2}} \\ $$$$\mathrm{11}\mid\mathrm{100}{a}+{b}\Rightarrow\mathrm{100}{a}+{b}=\mathrm{11}{p}^{\mathrm{2}} \Rightarrow\mathrm{11}\mid{a}+{b} \\ $$$$\left({a},{b}\right)\neq\left(\mathrm{6},\mathrm{5}\right),{b}\neq\mathrm{2},\mathrm{3},\mathrm{6},\mathrm{7},\mathrm{8}\left({a}\:{perfect}\:{square}\:{cannot}\right. \\ $$$$\left.{end}\:{in}\:\mathrm{2},\mathrm{3},\mathrm{7}\:{or}\:\mathrm{8},{power}\:{of}\:\mathrm{2}\:{in}\:\overset{\_\_\_\_} {{aabb}}\:{when}\:{b}\:=\:\mathrm{6}\:{is}\:\mathrm{1}\right) \\ $$$$\Rightarrow{Possible}\:{values}\:{of}\:\left({a},{b}\right)=\left(\mathrm{2},\mathrm{9}\right),\left(\mathrm{7},\mathrm{4}\right) \\ $$$${Of}\:{these},{only}\:\left(\mathrm{7},\mathrm{4}\right)\:{gives}\:{a}\:{perfect}\:{square} \\ $$$$\Rightarrow{N}=\mathrm{7744} \\ $$
Commented by BaliramKumar last updated on 27/May/23
$$\mathrm{perfect}\:\mathrm{square}\:\mathrm{number}\:\mathrm{last}\:\mathrm{two}\:\mathrm{same}\:\mathrm{digit}\:\mathrm{only} \\ $$$$\mathrm{00}\:\mathrm{or}\:\mathrm{44} \\ $$$$\because\:\mathrm{11}\mid\mathrm{a}+\mathrm{b} \\ $$$$\therefore\:\mathrm{bb}\:\neq\:\mathrm{00}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{if}\:\mathrm{bb}\:=\:\mathrm{00}\:\mathrm{then}\:\mathrm{11}=\:\mathrm{a}>\mathrm{9},\:\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{possible}\right) \\ $$
Commented by MM42 last updated on 28/May/23
$${bravo} \\ $$