<s>[INST] how can we get from $x \equiv 4 \pmod 9$ to $x \equiv 1 \pmod 3$ I want to solve the system:
$$ x \equiv 1 \pmod 3 , x \equiv 2 \pmod 5, x \equiv 3 \pmod 7, x \equiv 4 \pmod 9, x \equiv 5 \pmod {11}$$
The numbers $3,5,7,9,11$ are not pairwise coprime,especially the numbers $3,9$ are not coprime.
According to my notes, $x \equiv 4 \pmod 9 \Rightarrow x \equiv 4 \pmod 3 \Rightarrow x \equiv 1 \pmod 3$
But..how can we get from the relation $x \equiv 4 \pmod 9$ to this one: $x \equiv 1 \pmod 3$ ?
 [/INST] $x\equiv4(\text{mod }9)$ is equivalent to:
$x-4=9k (\text{ for some integer k})\implies x-3-1=3(3k)\implies x-1=3(3k+1)$
or, $x\equiv1(\text{mod }3)$
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