number prime to $bc$ with system of congruences Can you please help me to understand why all numbers $x$ prime to $bc$ are all the solutions of this system?
$$\begin{align*}
x&\equiv k\pmod{b}\\
x&\equiv t\pmod{c}
\end{align*}$$
Here $k$ is prime to $b$, and $t$ is prime to $c$.

Hint $\rm\quad \begin{eqnarray} x\equiv k\,\ (mod\ b)&\Rightarrow&\rm\:(x,b) = (k,b) = 1 \\
\rm x\equiv t\,\,\ (mod\ c)\,&\Rightarrow&\rm\:(x,c) =\, (t,c) = 1\end{eqnarray} \Bigg\}\ \Rightarrow\ (x,bc) = 1\ \ by\ Euclid's\ Lemma$
