Question

determinatii numerele de forma abc, mai mici decat 500, daca:
a) abc da restul 5 la impartirea cu 9
b) ( a+b+c ) si (abc + 2) sunt divizibile cu 7
Urgent!!​

Answer 1

$\it \overline{abc} =9k+5 \Rightarrow \overline{abc}-5=9k \Rightarrow \overline{abc}-5\in M_9 \ \ \ \ (1) \\ \\ \overline{abc}+2=7m \Rightarrow \overline{abc} =7m-2|_{-5} \Rightarrow\overline{abc}-5 =7m-7\Rightarrow \overline{abc}-5\in M_7 \ \ \ \ (2) \\ \\ (1),\ (2) \Rightarrow\overline{abc}-5\in M_9 \cap M_7 \Rightarrow \overline{abc}-5\in \{126,\ 189,\ 252,\ 315,\ 378,441\}|_{+5}\Rightarrow\\ \\ \overline{abc} \in\{131,\ 194,\ 257,\ 320,\ 383,\ 446\} \ \ \ \ \ (3)\\ \\ (a+b+c)\ \vdots\ 7\ \ \ \ \ (4)$

$\it (3),\ (4) \Rightarrow \overline{abc} \in\{194,\ 257,\ 383,\ 446\}$