Question

Calculeaza numarul natural din 3 cifre xyz cu bara deasupra stiind ca 26/x^2+y^2+z^2 este numar natural.

Answer 1

   
$\displaystyle\\\frac{26}{x^2+y^2+z^2}\in N\\\\ \text{Descompunem numarul 26 in suma de patrate perfecte: }\\\\26=1+9+16=1^2+3^2+4^2\\\\\frac{26}{1^2+3^2+4^2}=\frac{26}{26}=1\in N\\\\ \Longrightarrow~\boxed{x = 1;~y=3;~z=4 }\\\\\Longrightarrow~\overline{xyz}=\boxed{134}\\\\\text{Datorita simetriei necunoscutele pot comuta intre ele.}\\\\\Longrightarrow~Avem~3!~solutii=1\times2\times3=6~solutii.~~("!" = "factorial")\\\\\Longrightarrow~\boxed{\overline{xyz}\in \{134;~143;~314;~341;~413;~431 \}}$