Question

1.Gasiti toate numerele naturale cuprinse intre 13 si 32 care au printe divizori cel putin doua patrate perfecte.2.Cati multiplii impari ai numarului 7 sunt cuprinsi intre 101 si 303.

Answer 1

$1) \\ 13<x<32;~x~\vdots~p^2~s~iq^2 \\ Pentru~p^2=1~si~q^=4: \\ x \in \{16,20,24,28\} \\ Pentru~p^2=1~si~q^2=9: \\ x \in \{18,27\} \\ Pentru~p^2=1~si~q^2=16: \\ x \in \{16\} \\ Pentru~p^2=4~si~q^2=9: \\ x \in \varnothing \\ Pentru~p^2=4~si~q^2=16: \\ x \in \{16\} \\ Pentru~p^2=9~si~q^2=16: \\ x \in \varnothing \\ \boxed{Numere:16,18,20,24,27,28}$


$2) \\ 101<M_{7}<303 \\ M_7 \in \{105,112,119,.......287,294,301\} \\ card(M_7)=30 \\ Observam~ca~sunt~din~2~in~2~impari \Longrightarrow \boxed{card(M_{7~impari})=15}$