Modular homework problem Show that:
$$[6]_{21}X=[15]_{21}$$
I'm stuck on this problem and I have no clue how to solve it at all.

Well, we know that $\gcd\ (6,21)=3$ which divides $15$. So there will be solutions:
$$
\begin{align}
6x &\equiv 15 \pmod {21} \\
2x &\equiv 5 \pmod 7
\end{align}
$$
because that $2\times 4\equiv 1 \pmod 7$, thus:
$$
\begin{align}
x &\equiv 4\times 5 \pmod 7\\
 &\equiv 6 \pmod 7
\end{align}
$$
