I'm trying to learn how to use the Chinese Remainder Theorem (CRT), and in order to give some context:
We search for all $x ∈ Z$, where $Z$ is the set of integers.
The easy case (which I can solve) is if all $m_i$, where $i=1,2,...,k$ are pairwise coprime.
$x≡5\pmod 6 $
Then the first equation is satisfied iff $x=4+5s$, for some $s ∈ Z$.
These $x$ also satisfy the second equation iff $4+5s≡_6 5 ↔ -s≡_6 1 ↔ s=-1+6t$, for some $t ∈ Z$. Thus $x=4+5(-1+6t)=-1+30t$.
Lastly, these $x$ also satisfy the third equation iff $-1+30t ≡_7 3 ↔ 2t ≡_7 4 ↔ t ≡_7 2 ↔ t = 2+7n$, for some $n ∈ Z$. Thus $x=59+210n$.
Now to my issue, I have the problem:
Here $\gcd(4,6)=2$, so they are not coprime and I don't know how to solve this. Can someone please solve it and explain why the problem becomes more difficult to solve when $m_i$ are not pairwise coprime.