What is the property of co-primes that allows CRT to work?

Question

I have been reading about the Chinese Remainder Theorem and I have the following question:

Basically the CRT says that there is a $1$ to $1$ correspondance between a number $N \in [0, m\cdot n)$ and pairs $(a,b)\space s.t\space 0 \le a \lt m \space \And \space 0 \le b \lt n $ and $\gcd(m,n) = 1$

I have checked with numbers that are not relatively prime and indeed there is no $1$ to $1$ correspondence e.g. if $m = 2$ and $n = 4$ so $gcd(2,4) = 2$ and $m\cdot n = 8$
we have:
$3 \equiv [1,3] \mod [2,4]$
$7 \equiv [1,3] \mod [2,4]$

What I am interested in is, what exactly is the property that guarantees the $1-1$ correspondence in the case of co-prime $m,n$? Or conversely how exactly using non coprime modulo messes up the CRT?

Answer 1

Basically CRT is simply the observations that:

• LCM of moduli is the length where all cycles repeat again from the start in sync.
• GCD being 1, implies LCM is the same as the product.
• if $y=mx+b=nz+a$ has a solution, it is before $z=m$ or $x=n$
• This solution is unique, up to translation.