Question

abc + bc + c=444
aflati abc
Toate sunt in baza zece

Answer 1

$\it \overline{abc}+\overline{bc}+c=444 \Rightarrow 100a+10b+c+10b+c+c=444 \Rightarrow \\ \\ \Rightarrow100a+20b+3c=444 \Rightarrow 100a+10b=444-3c\ \ \ (1)\\ \\ \\ 100a+10b \in M_{10}\ \ \ (2)\\ \\ (1), (2) \Rightarrow 444-3c\in M_{10} \Rightarrow u(3c)=4 \Rightarrow 3c=24 \Rightarrow c=8\ \ \ (3)\\ \\ (2),\ (3) \Rightarrow 100a+20b=420|_{:20} \Rightarrow 5a+b=21\ \ \ \ (4)$

$\it (4) \Rightarrow \begin{cases}\it a=4,\ b=1\\ \\ \it a=3,\ b=6\end{cases}\\ \\ \\ Deci,\ \overline{abc} \in \{368,\ \ 418\}$