Question

Determinați numere ab pentru care ab +ba este pătrat perfect​

Answer 1

$\overline{ab}+\overline{ba}=10a+b+10b+a=11(a+b) \\ \\ 11(a+b)~este~patrat~perfect.\\ \\ Vlaoarea~minima~a~sumei~a+b~este~2~(a,b\in\{1;2;3;4;5;6;7;8;9\})\\ \\Iar~valoarea~maxima~este~18\\ \\ 11\cdot 2=22~si~11\cdot 18=198\\ \\11(a+b)=11\cdot n^2 \cdot 11~(a+b=11\cdot n^2,~unde~n\in \mathbb{N}).\\ \\ Patratele~perfect~cuprinse~intre~2~si~198,~care~sunt~multiple~de~11~sunt:~121\\ \\ \Rightarrow 11(a+b)=121 \Rightarrow a+b=11\\ \\ OBS:~a,b\in \{1; 2; 3; 4;5;6;7;8;9\}\\ \\ S=\{(2,~9);~(3,~8);~(4,~7);~(5,~6);~(6,~5);~(7,~4);~(8,~3);~(9,~2)\}$