Question

xx+yx+zx=xy3. Aflati cifrele distincte

Answer 1

   
[tex]\bf \overline{xx}+\overline{yx}+\overline{zx}=\overline{xy3}\\\\ \texttt{ x, y si z sunt cifre.}\\ \texttt{Calculam suma cifrelor de la unitati:}\\\\ x+x+x=3\cdot x=3~~sau=13~~sau=23~~sau=33\\ 3\cdot x \texttt{ nu poate fi egal cu 13 sau 23 deoarece}\\ \texttt{ 13 si 23 sunt numere prime, }\\\\ 3\cdot x \texttt{ nu poate fi egal cu 33 deoarece}\\ 33 = 3\cdot 11\texttt{, iar x nu poate fi egal cu 11 }\\ \texttt{deoarece x este cifra.}\\\\ \Rightarrow~x=1 \Rightarrow~3\cdot x = 3\cdot1=3 [/tex]


[tex]\dysplaystyle \bf \\ \texttt{Calculam suma cifrelor de la zeci:}\\\\ x+y+z = 10\cdot x + y,~~~dar~stim ~ca~~x = 1 \\\\ 1+y+z = 10\cdot 1 + y\\\\ 1+y+z = 10 + y\\\\ 1+\underline{y}+z -\underline{y} = 10\\\\ 1+z=10\\\\ z = 10-1 = 9\\\\ \texttt{y poate lua orice valoare de la 2 pana la 8.}\\\\ \texttt{Solutia:}\\\\ \boxed{ \left\{\begin{array}{c}\bf x=1~~~~~~~~~~~~~~~~~~~~~\\\bf y\in\{2;3;4;5;6;7;8\}\\\bf z=9~~~~~~~~~~~~~~~~~~~~~\end{array}\right } [/tex]