Solving a linear congruence with no common factor?

Question

$$9x \equiv 7 \pmod{1125}$$

I'm not sure how to start reducing since 9 and 7 don't have any common factors. Does the Chinese Remainder Theorem apply here?

Answer 1

Since $1125 = 9\cdot 125$ this congruence has no solution. Why?

Because we have $$9\mid 9x-7\implies 9\mid 7$$

A contradiction.

Answer 2

If $9x \equiv 7 \bmod 1125$, then $0 \equiv 1125 x = 125 \cdot 9 x \equiv 125 \cdot 7 = 875 \not\equiv 0 \bmod 1125$, a contradiction.