Chinese Remainder Theorem quiz question not right?

Question

Why is answer 2 not right?

To apply Chinese Remainder Theorem to solve a system of modular equations all the moduli have to be pairwise relatively prime. Given the following system of modular equations:

$x\equiv 3\pmod 3$

$x\equiv 5\pmod 9$

$x\equiv 4\pmod 5$

we should only select the following in order to apply Chinese Remainder Theorem:

$x\equiv 5\pmod 9$

$x\equiv 4\pmod 5$.

Why is this the case? I think $x\equiv 3\pmod 3$ and $x\equiv 4 \pmod5$ should also be right since you can't fully solve the system of modular equations either way.

Answer 1

@alramdkwan, first, welcome to MSE.

Now, concerning your question, I think that it already implies that there is no solution exists because:

1. $x \equiv 3 \pmod{3}$ means $x$ is divisible by 3
2. $x \equiv 5 \pmod{9}$ means $x$ is not divisible by 3

The two conditions above yield the result of no $x$,

About your solution, I think it is true that in two cases you have two answers. But this is not incorrect. It rather leads you to the conclusion that this system has no solution.