### Instruction: What is the sum of all positive integers $r$ that satisfy $$\mathop{\text{lcm}}[r,700] = 7000~?$$.### Response: To solve this problem, I need to find all the possible values of $r$ that make the least common multiple of $r$ and $700$ equal to $7000$.
I know that the least common multiple of two numbers is the smallest positive integer that is a multiple of both numbers.
I also know that the least common multiple of two numbers can be found by multiplying them together and dividing by their greatest common divisor.
So, I can write an equation for the least common multiple of $r$ and $700$ as follows: $$\mathop{\text{lcm}}[r,700] = \frac{r \times 700}{\mathop{\text{gcd}}[r,700]}$$
I want this expression to be equal to $7000$, so I can set it equal to $7000$ and cross-multiply to get: $$r \times 700 = 7000 \times \mathop{\text{gcd}}[r,700]$$
I can simplify this equation by dividing both sides by $700$, which gives me: $$r = 10 \times \mathop{\text{gcd}}[r,700]$$
Now, I have a simpler equation that relates $r$ and the greatest common divisor of $r$ and $700$.
I can use this equation to find the possible values of $r$.
I notice that $r$ must be a multiple of $10$, since it is equal to $10$ times something.
So, I can write $r$ as $10k$, where $k$ is some positive integer.
Substituting $r = 10k$ into the equation, I get: $$10k = 10 \times \mathop{\text{gcd}}[10k,700]$$
I can divide both sides by $10$, which gives me: $$k = \mathop{\text{gcd}}[10k,700]$$
Now, I have an equation that relates $k$ and the greatest common divisor of $10k$ and $700$.
I can use this equation to find the possible values of $k$.
I notice that $k$ must be a divisor of $700$, since it is equal to the greatest common divisor of $10k$ and $700$.
So, I can list the divisors of $700$ as follows: $1, 2, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, 700$.
For each of these divisors, I can plug it into the equation for $k$ and check if it satisfies the equation.
For example, if $k = 1$, then the equation becomes $1 = \mathop{\text{gcd}}[10,700]$, which is true, since the greatest common divisor of $10$ and $700$ is $10$.
So, $k = 1$ is a solution, and that means $r = 10k = 10$ is a solution as well.
Similarly, if $k = 2$, then the equation becomes $2 = \mathop{\text{gcd}}[20,700]$, which is also true, since the greatest common divisor of $20$ and $700$ is $20$.
So, $k = 2$ is a solution, and that means $r = 10k = 20$ is a solution as well.
I can continue this process for the other divisors of $700$ and find that the only ones that satisfy the equation for $k$ are $k = 1, 2, 5, 7, 10, 35, 70$.
These correspond to the values of $r = 10k$ as $r = 10, 20, 50, 70, 100, 350, 700$.
These are all the possible values of $r$ that make the least common multiple of $r$ and $700$ equal to $7000$.
To find the sum of all these values, I can add them up as follows: $10 + 20 + 50 + 70 + 100 + 350 + 700 = 1300$.