Question

Determinati restul impartirii la 1000 a numarului impar abcd(barat) pentru care $\sqrt{5 \sqrt{abcd} }$ ∈ N.

Answer 1

   
[tex]\text{Trebuie sa aflam numarul }~~ \overline{abcd}. \\ \\ \text{Stim ca: }~~\sqrt{5 \sqrt{~\overline{abcd}~} } \in N \\ \\ \Longrightarrow~~~ \sqrt{~\overline{abcd}~} \in N \\ \\ \Longrightarrow~~~\overline{abcd} ~~~\text{este patrat perfect.} \\ \\ \text{Cautam patratele perfecte de 4 cifre: } \\ \\ \sqrt{1000}= 31,62 \\ \sqrt{9999}= 99,99 \\ \\ \Longrightarrow~~~\text{Patratele perfecte de 4 cifre sunt:} \\ \{32^2;~33^2;~34^2;~35^2;~36^2;~\cdots ;~99^2\} [/tex]


[tex]\Longrightarrow ~~~ \sqrt{~\overline{abcd}} \in \{32;~33;~34;~35;~36;~\cdots ;~99\} \\ \\ \text{Dar }~~\sqrt{5 \sqrt{~\overline{abcd}~} } \in N \\ \\ \Longrightarrow ~~~5 \sqrt{~\overline{abcd}~} ~~~\text{este patrat perfect.} \\ \\ \Longrightarrow~~~\text{Din multimea: } \{32;~33;~34;~35;~36;~\cdots ;~99\} \\ \text{alegem numerele pe care le putem descompune in doi factori, }\\ \text{din care unul este 5 iar celalalt este patrat perfect.} [/tex]


[tex]\text{Aceste numere sunt: } \\ 5 \times 3^2 = 45\\ 5 \times 4^2 = 80 \\\\ \Longrightarrow~~~\texttt{Avem doua solutii: }\\\\ \texttt{Solutia 1}:\\ \overline{abcd}= 45^2 = 2025 \\ \\ 2025 : 1000 = 2 ~\texttt{rest}~ \boxed{25}\\ \\ \text{Verificam: } ~~ \sqrt{5 \sqrt{~\overline{2025} ~}}= \sqrt{5\times 45~}=\sqrt{225}= 15 \in N \\ \\ \\ [/tex]


[tex]\texttt{Solutia 2}:\\ \overline{abcd}= 80^2 = 6400 \\ \\ 6400 : 1000 = 6 ~\texttt{rest}~ \boxed{400}\\ \\ \text{Verificam: } ~~ \sqrt{5 \sqrt{~\overline{6400} ~}}= \sqrt{5\times 80~}=\sqrt{400}= 20 \in N [/tex]


Nota:
Solutia 2 va fi eliminata deoarece 6400 nu este numar impar, asa cum cere enuntul.