The answer is $\boxed{x\equiv 557\pmod{1540}}$. This is a standard mod equation and thus has a standard solution. Please see the book  Number Theory (MOD) . $$\begin{array}{} 17x &\equiv 229 &\pmod{1540} &\implies 17x &= 229 + 1540y \\ & & &\implies 0 &\equiv 8 + 10y &\pmod{17}\\ & & &\implies 7y &\equiv 8 &\pmod{17} \end{array}$$ $$\begin{array}{} \therefore\quad 7y &= 8 + 17u & &\implies 0 &\equiv 1 + 3u&\pmod{7}\\ &&&\implies 3u &\equiv -1 &\pmod{7} \\ &&&\implies u&\equiv 2 &\pmod{7} \end{array}$$ Therefore: $$\begin{array}{} y\equiv \frac{8+17\times 2}{7}&\pmod{17} &\implies y \equiv 6&\pmod{17}\\ x\equiv \frac{229 + 1540\times 6}{1540}&\pmod{1540} &\implies x \equiv 557&\pmod{1540} \end{array}$$