Question

Numerele 1507,364,458 inpartite la acelasi nr natural x dau respectiv resturile 7,4,8.
Determinați x .

Answer 1

[tex] Din~teorema~impartirii~cu~rest,~obtinem: \\ \\ 1507= c_{1}*x+7 \\ 364= c_{2}*x+6 \\ 458= c_{3} *x+8 \\ \\ (x\ \textgreater \ 8;~impartitorul~este~mai~mare~decat~catul) \\ \\ Deci: \\ \\ 1500= c_{1}*x \\ 360= c_{2}*x \\ 450 =c_{3} *x $[tex]Din~ultimele~3~relatii,~obtinem~ca~x~este~un~divizor ~comun~al~nu- \\ merelor~1500,~360~si~450. \\ \\ 1500=2 ^{2}*3*5^3 \\ 360=2^3*3^2*5 \\ 450=2*3^2*5^2$ $(1500;360;450)=2*3*5=30. \\ \\ Asadar,~x \in D_{30},~si~cum~x\ \textgreater \ 8 \Rightarrow \boxed{x \in \{10;15;30\}} \leftarrow Solutia.$

Answer 2

$1507:x=c_1 \ rest \ 7 \\\\ 1507=xc_1+7 \ \ |-7 \\\\ 1500=xc_1 \\\\\\ 364:x=c_2 \ rest \ 4 \\\\ 364=xc_2+4 \ \ \ |-4 \\\\ 360=xc_2 \\\\\\ 458:x=c_3 \ rest \ 8 \\\\ 458=xc_3+8 \ \ \ |-8 \\\\ 450=xc_3 \\\\\ c_1=\frac{1500}{x} \\\\\ c_2=\frac{360}{x} \\\\ c_3=\frac{450}{3} \\\\ (1500;360;450)=30 \\\\ x\in D_{30} \\\\ D=I*C+R \ \ \ ( I\ \textgreater \ R) \\\\ \boxed{(x\ \textgreater \ 8)} \\\\\\D_{30} \vee \ x \ \textgreater \ 8 \Longrightarrow \boxed{\boxed{\{10;15;30\}}}$