What is the sum of all positive integers $r$ that satisfy $$\mathop{\text{lcm}}[r,700] = 7000~?$$
Note the prime factorizations $700=2^2\cdot 5^2\cdot 7$ and $7000=2^3\cdot 5^3\cdot 7$.

If $\mathop{\text{lcm}}[r,700]=7000$, then in particular, $r$ is a divisor of $7000$, so we can write $r=2^\alpha\cdot 5^\beta\cdot 7^\gamma$, where $0\le\alpha\le 3$, $0\le\beta\le 3$, and $0\le\gamma\le 1$.

Moreover, we know that $\mathop{\text{lcm}}[r,700]=2^{\max\{\alpha,2\}}\cdot 5^{\max\{\beta,2\}}\cdot 7^{\max\{\gamma,1\}}$, and we know that this is equal to $7000=2^3\cdot 5^3\cdot 7$. This is possible only if $\alpha=3$ and $\beta=3$, but $\gamma$ can be $0$ or $1$, giving us two choices for $r$: $$r = 2^3\cdot 5^3\cdot 7^0 = 1000 \text{~~or~~} r=2^3\cdot 5^3\cdot 7^1 = 7000.$$So the sum of all solutions is $1000+7000=8000$.