If $r$ is a positive integer such that $\mathop{\text{lcm}}[r,100]\cdot\gcd(r,100) = 13200$, then what is $\mathop{\text{lcm}}[r,100]$?
I know that the product of the least common multiple and the greatest common divisor of two numbers is equal to the product of the numbers themselves, by a well-known identity. So, I can rewrite the given equation as $r\cdot 100 = 13200$. Dividing both sides by 100, I get $r = 132$. Now, to find the least common multiple of 132 and 100, I can use the formula $\mathop{\text{lcm}}[a,b] = \frac{ab}{\gcd(a,b)}$. The greatest common divisor of 132 and 100 is 4, since 4 is the largest number that divides both of them evenly. So, $\mathop{\text{lcm}}[r,100] = \frac{r\cdot 100}{\gcd(r,100)} = \frac{132\cdot 100}{4} = 3300$. 