Does this congruence equation system have no solutions?

Question

$$ \left\{ \begin{array}{c} 9x \equiv 5 \pmod{10} \\ 14x \equiv 8 \pmod{18} \\ \end{array} \right. $$

$m.c.m.(10,18)=2$

$2 \nmid (5+8)$

There are no integer solutions for this system?

Answer 1

For CRT there are integer solutions for this system, indeed

$$\left\{ \begin{array}{c} 9x ≡ 5 \pmod {10} \\ 14x ≡ 8 \pmod {18} \\ \end{array} \right. \iff \left\{ \begin{array}{c} 9x ≡ 5 \pmod {2} \\ 9x ≡ 5 \pmod {5} \\ 14x ≡ 8 \pmod {9} \\ 14x ≡ 8 \pmod {2} \\ \end{array} \right.\iff \left\{ \begin{array}{c} x ≡ 1 \pmod {2} \\ 4x ≡ 0 \pmod {5} \\ 5x ≡ -1 \pmod {9} \\ 0 ≡ 0 \pmod {2} \\ \end{array} \right.\iff \left\{ \begin{array}{c} x ≡ 1 \pmod {2} \\ x ≡ 0 \pmod {5} \\ 7\cdot 5x ≡ -7 \pmod {9} \\ \end{array} \right.\iff\left\{ \begin{array}{c} x ≡ 1 \pmod {2} \\ x ≡ 0 \pmod {5} \\ x ≡ 7\pmod {9} \\ \end{array} \right.$$

which solutions are $$x\equiv 25 \pmod {90} \iff x=25+90k$$