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: $x \equiv 1 \pmod 3$ ?

Since 3 is a factor of 9, $x \equiv_9 4 \Rightarrow x \equiv_3 4 \Rightarrow x \equiv_3 1$
