Question

Determinați toate nr naturale de forma ab care au proprietatea ca 9
_____
a^(2)+b^(2)
apartine nr nat

Answer 1

$\frac{9}{a^2+b^2}\in\mathbb{N}\Rightarrow a^2+b^2\mid9 \\ \\ a^2\in\{1,3,9\}$
$1)~a^2+b^2=1 \\ \\ 1=0+1=1+0 \\ \\ (a^2,~b^2)\in\{(0,1),(1,0)\} \\ \\ (a,b)\in\{(0,1),(1,0)\}$
$2)~a^2+b^2=3 \\ \\ 3=0+3=1+2=2+1=3+0 \\ \\ 3~nu~poate~fi~scris~ca~suma~de~doua~patrate~perfecte$
$3)~a^2+b^2=9 \\ 9=0+9=9+0 \\ \\ (a^2,~b^2)\in\{(0,9),(9,0)\} \\ \\ (a,b)\in\{(0,3),(3,0)\}$
$(a,b)\in\{(0,1),(1,0),(0,3),(3,0)\} \\ \\ Dar~a\neq0 \\ \\ \bar{ab}\in\{10,30\}$