Can someone grade my proof for this problem:
Let $n$ be a positive integer. Prove that there exists a positive integer $m > 1000$ with the following two properties: $m$'s last 3 digits are 007, and $m$ is relatively prime to $n$.
Proof: Consider $2$ cases, $\gcd(n, 1000) = 1$, and $\gcd(n, 1000) > 1$.
Case 1: $\gcd (n, 1000) = 1$. Let us choose an $m$ such that $$m \equiv 1 \pmod{n}.$$ We also know that $m \equiv 7 \pmod{1000}$. By the Chinese Remainder Theorem (CRT), we can guarantee the existence of such an $m$ since $n$ and $1000$ are relatively prime.
Case 2: $\gcd(n, 1000) > 1$. This case implies $n$ is divisible by $2$ or $5$. Thus we let rewrite $n$ as $n=2^x 5^y k$, where $x, y > 0$, and $\gcd(k, 1000) = 1$. Note that it suffices to show that $m$ is coprime to $k$, since $m$ is already coprime to $2$ and $5$ (ends in $007$). We can choose an $m$ that is congruent to $1 \pmod k$. We have the system $$\begin{cases} m\equiv 7 \pmod{1000} & \\ m\equiv 1 \pmod{k} \end{cases}.$$ By CRT, we are guaranteed a solution for such an $m$, and thus we are done. $\blacksquare$