Question

Aratati ca numarul a=3^21+3^20+3^19)÷39 este patrat perfect.


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Answer 1

$a=\\=\frac{3^{21}+3^{20}+3^{19}}{39}=\\=\frac{3^{2+19}+3^{1+19}+3^{0+19}}{3*13}=\\ =\frac{3^{2}*3^{19}+3^{1}*3^{19}+3^{0}*3^{19}}{3*13}=\\=\frac{(3^{2}+3^{1}+3^{0})*3^{19}}{3*13}=\\= \frac{(9+3+1)*3^{19}}{3*13}=\\=\frac{(12+1)*3^{19}}{3*13}=\\=\frac{13*3^{19}}{3*13}=\\=\frac{3^{19}}{3}=\\=\frac{3^{19}}{3^{1}}=\\=3^{19-1}=\\=3^{18}=\\=3^{9*2}=\\=(3^{9})^{2} \to patrat.perfect$