Suppose $x$ is an integer such that $3x \equiv 15 \pmod{64}$. Find remainder when $q$ is divided by $64$. Suppose $x$ is an integer such that $3x \equiv 15 \pmod{64}$. If $x$ has remainder $2$ and quotient $q$ when divided by $23$, determine the remainder when $q$ is divided by $64$.
I tried a couple things. By division algorithm, know $x = 23q + 2$. So $3(23q + 2) \equiv 15 \pmod{64}$. Not sure how to go from there.
Another thing I tried is $3x = 64q + 15$. If we let $q$ be zero, then $x$ is obviously $5$. This also doesn't wind up being that helpful, and I think $x$ can have other values asides from $5$.

Continuing from where you stopped:
$$\begin{align} 3(23q + 2) &\equiv 15 \pmod{64} \Rightarrow \\
69q+6 &\equiv 15 \pmod{64} \Rightarrow \\
69q &\equiv 9 \pmod{64} \Rightarrow \\
64q+5q &\equiv 9 \pmod{64} \Rightarrow \\
5q &\equiv 9 \pmod{64} \Rightarrow \\
5q\cdot 13 &\equiv 9\cdot 13 \pmod{64} \Rightarrow \\
65q &\equiv 117 \pmod{64} \Rightarrow \\ 
64q+q &\equiv 64+53 \pmod{64} \Rightarrow \\ 
q &\equiv 53 \pmod{64}.
\end{align}$$
