Question

a=1x2x3x4x......x n+ 57 a) daca n= 10, aratati ca nu este patrat perfect. b)determinati numarul n pentru care a este patrat perfect

Answer 1

$\displaystyle a)~1 \cdot 2 \cdot 3 \cdot ... \cdot 10~are~ultima~cifra~0,~deci~A~are~ultima~cifra~7. \\ \\ Rezulta~ca~A~nu~este~patrat~perfect. \\ \\ Reamintesc~ca~un~patrat~perfect~nu~poate~avea~ultima~cifra~\\ \\ 2,3,7~sau~8. \\ \\ b)~Pentru~n \ge 5,~in~produsul~1 \cdot 2 \cdot 3 \cdot... \cdot n~apar~factorii~2~si~5,~deci \\ \\ numarul~1 \cdot 2 \cdot 3 \cdot ...n~se~termina~in~0,~si~deci~A~se~termina~in~7, \\ \\ prin~urmare~nu~este~patrat~perfect.$

$\displaystyle Ramane~sa~verificam~variantele~n \in \{1,2,3,4\}. \\ \\ n=1 \Rightarrow A=58,~nu~convine \\ \\ n=2 \Rightarrow A=59,~nu~convine \\ \\ n=3 \Rightarrow A=63,~nu~convine \\ \\ n=4 \Rightarrow A=81,~convine! \\ \\ Raspuns:~n=4.$