Question

afla nr de forma ab care prin impartire la suma cifrelor lor dau citul mai mic decit 10

Answer 1

    
[tex]\displaystyle \\ \frac{\overline{ab}}{a+b} \ \textless \ 10 ~~~unde ~~a\ \textgreater \ 0 \\ \\ \frac{10a+b}{a+b} \ \textless \ 10 \\ \\ \frac{10a+b}{a+b} = \frac{a+b+9a}{a+b}= \frac{a+b}{a+b}+\frac{9a}{a+b} =1+\frac{9a}{a+b} \\ \\ Daca ~b = 0~~\Longrightarrow ~~1+\frac{9a}{a+b}=1+\frac{9a}{a+0}=1+\frac{9a}{a} =1+9=10\\ \\ Daca ~b \ \textgreater \ 0~~\Longrightarrow ~~\frac{9a}{a+b}\ \textless \ 9 ~~\Longrightarrow~~ 1+\frac{9a}{a+b}\ \textless \ 10 [/tex]


[tex]\displaystyle \Longrightarrow ~~\frac{\overline{ab}}{a+b} \ \textless \ 10 ~~daca:\\ \\ \boxed{\overline{ab} \in \{11;~12;~13;~14;~\hdots 99\} - \{10;~20;~30;~ \hdots 90\}} [/tex]