Question

Aflati suma cifrelor numarului abc in baza 10 ; stiind ca abc in baza 10=ab in baza 10+bc in baza 10+ca in baza 10

Answer 1

$\overbar{abc}=\overbar{ab}+\overbar{bc}+\overbar{ca}\\</p><p>\\ 100a+10b+c=10a+b+10b+c+10c+a\\</p><p>\\ 100a+10b+c=11a+11b+11c~|-11a-10b-c\\</p><p>\\ 89a=b+10c\\</p><p>\\89a=\overbar{bc}\\</p><p>\\ 10 \leq \overbar{bc} \leq 99\\</p><p>\\ (1 \leq a \leq 9~|\cdot 89)\\</p><p>\\89 \leq 89a \leq 801\\</p><p>\\ In~acest~caz~89a=\{89,~178,~...,~801/}~Dar~89a~este~egal~cu~un~nr~de~doua~cifre~(bc)\\</p><p>\\ \Rightarrow 89a=\overbar{bc}~<=>\\ </p><p>\\ <=>~a=1,~si~\overbar{bc}=89 \Rightarrow b=8,~c=9$

Raspuns: a+b+c=1+8+9=18