Решите в натуральных числах уравнение: x+6y=50.

\[x + 6y = 50;\ \ \ (2;8) -\]

\[частное\ решение.\]

\[x = 2 + 6n;\ \ \ y = 8 - n;\ \ n \in Z.\]

\[Для\ x:\ \ n \geq 0;\ \ для\ y:\ \ \]

\[n \leq 7 \Longrightarrow n = 0;1;2;3;4;5;6;7.\]

\[n = 0:\]

\[x = 2 + 6 \bullet 0 = 2;\ \ y = 8 - 0 =\]

\[= 8.\]

\[n = 1:\]

\[x = 2 + 6 \bullet 1 = 8;\ \ \ y = 8 - 1 =\]

\[= 7.\]

\[n = 2:\]

\[x = 2 + 6 \bullet 2 = 2 + 12 = 14;\ \ \ \]

\[\ \ y = 8 - 2 = 6.\]

\[n = 3:\]

\[x = 2 + 6 \bullet 3 = 2 + 18 = 20;\ \]

\[\ \ y = 8 - 3 = 5.\]

\[n = 4:\]

\[x = 2 + 6 \bullet 4 = 2 + 24 = 26;\]

\[\ y = 8 - 4 = 4.\]

\[n = 5:\]

\[x = 2 + 6 \bullet 5 = 2 + 30 = 32;\ \]

\[\ y = 8 - 5 = 3.\]

\[n = 6:\]

\[x = 2 + 6 \bullet 6 = 2 + 36 = 38;\ \ \]

\[\ \ y = 8 - 6 = 2.\]

\[n = 7:\]

\[x = 2 + 6 \bullet 7 = 2 + 42 = 44;\ \]

\[\ \ y = 8 - 7 = 1.\]

\[Ответ:x = 2,\ y = 8;\ \ x = 8,\ \]

\[y = 7;\ \ x = 14,\ y = 6;\ \ x = 20,\]

\[\ y = 5;\]

\[\ \ \ \ \ \ \ \ \ \ \ \ \ x = 26,\ y = 4;\ \ \ x = 32,\ \]

\[y = 3;\ x = 38,\ y = 2;\ \ \ x = 44,\]

\[\ y = 1.\]