Question

aratati ca numarul b este patrat perfect, unde: b=0,(19)×(1+2+3+...+2013):20intregi1/3×1/53

Answer 1

$b=0,(19)\cdot (1+2+3+...+2013):20\frac{1}{3}\cdot \frac{1}{53}\\</p><p>\\ b=\frac{19}{99}\cdot \frac {2013\cdot 2014}{2}:\frac{20\cdot 3+1}{3}\cdot \frac{1}{53}\\</p><p>\\ b=\frac{19}{99} \cdot 2013 \cdot 1007 \cdot \frac{3}{61}\cdot \frac{1}{53}\\</p><p>\\ b=\frac{19\cdot 2013 \cdot 1007 \cdot 3}{99\cdot 61 \cdot 53}\\</p><p>\\ b=\frac{19\cdot 2013 \cdot 19 \cdot 3}{99 \cdot 61}\\</p><p>\\ b=\frac{19^2 \cdot 6039}{6039}\\</p><p>\\ b=19^2 \Rightarrow b~este~patrat~perfect$